chapter 10 resource masters - nhv regional hs district | north
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Chapter 10Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 10 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828013-3 Algebra 2Chapter 10 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 10-1Study Guide and Intervention . . . . . . . . 573–574Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 575Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 576Reading to Learn Mathematics . . . . . . . . . . 577Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 578
Lesson 10-2Study Guide and Intervention . . . . . . . . 579–580Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 581Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 582Reading to Learn Mathematics . . . . . . . . . . 583Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 584
Lesson 10-3Study Guide and Intervention . . . . . . . . 585–586Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 587Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 588Reading to Learn Mathematics . . . . . . . . . . 589Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 590
Lesson 10-4Study Guide and Intervention . . . . . . . . 591–592Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 593Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 594Reading to Learn Mathematics . . . . . . . . . . 595Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 596
Lesson 10-5Study Guide and Intervention . . . . . . . . 597–598Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 599Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 600Reading to Learn Mathematics . . . . . . . . . . 601Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 602
Lesson 10-6Study Guide and Intervention . . . . . . . . 603–604Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 605Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 606Reading to Learn Mathematics . . . . . . . . . . 607Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 608
Chapter 10 AssessmentChapter 10 Test, Form 1 . . . . . . . . . . . 609–610Chapter 10 Test, Form 2A . . . . . . . . . . 611–612Chapter 10 Test, Form 2B . . . . . . . . . . 613–614Chapter 10 Test, Form 2C . . . . . . . . . . 615–616Chapter 10 Test, Form 2D . . . . . . . . . . 617–618Chapter 10 Test, Form 3 . . . . . . . . . . . 619–620Chapter 10 Open-Ended Assessment . . . . . 621Chapter 10 Vocabulary Test/Review . . . . . . 622Chapter 10 Quizzes 1 & 2 . . . . . . . . . . . . . . 623Chapter 10 Quizzes 3 & 4 . . . . . . . . . . . . . . 624Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 625Chapter 10 Cumulative Review . . . . . . . . . . 626Chapter 10 Standardized Test Practice . 627–628Unit 3 Test/Review (Ch. 8–10) . . . . . . . 629–630
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 10 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 10 Resource Masters includes the core materialsneeded for Chapter 10. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 10Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 572–573. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
Change of Base Formula
common logarithm
LAW·guh·RIH·thuhm
exponential decay
EHK·spuh·NEHN·chuhl
exponential equation
exponential function
exponential growth
exponential inequality
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
logarithm
logarithmic function
LAW·guh·RIHTH·mihk
natural base, e
natural base exponential function
natural logarithm
natural logarithmic function
rate of decay
rate of growth
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Study Guide and InterventionExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 573 Glencoe Algebra 2
Less
on
10-
1
Exponential Functions An exponential function has the form y � abx,where a � 0, b � 0, and b � 1.
1. The function is continuous and one-to-one.
Properties of an2. The domain is the set of all real numbers.
Exponential Function3. The x-axis is the asymptote of the graph.4. The range is the set of all positive numbers if a � 0 and all negative numbers if a � 0.5. The graph contains the point (0, a).
Exponential Growth If a � 0 and b � 1, the function y � abx represents exponential growth.and Decay If a � 0 and 0 � b � 1, the function y � abx represents exponential decay.
Sketch the graph of y � 0.1(4)x. Then state the function’s domain and range.Make a table of values. Connect the points to form a smooth curve.
The domain of the function is all real numbers, while the range is the set of all positive real numbers.
Determine whether each function represents exponential growth or decay.a. y � 0.5(2)x b. y � �2.8(2)x c. y � 1.1(0.5)x
exponential growth, neither, since �2.8, exponential decay, sincesince the base, 2, is the value of a is less the base, 0.5, is betweengreater than 1 than 0. 0 and 1
Sketch the graph of each function. Then state the function’s domain and range.
1. y � 3(2)x 2. y � �2� �x
3. y � 0.25(5)x
Domain: all real Domain: all real Domain: all real numbers; Range: all numbers; Range: all numbers; Range: allpositive real numbers negative real numbers positive real numbers
Determine whether each function represents exponential growth or decay.
4. y � 0.3(1.2)x growth 5. y � �5� �x
neither 6. y � 3(10)�x decay4�5
x
y
O
x
y
O
x
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1�4
x �1 0 1 2 3
y 0.025 0.1 0.4 1.6 6.4
x
y
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Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 574 Glencoe Algebra 2
Exponential Equations and Inequalities All the properties of rational exponentsthat you know also apply to real exponents. Remember that am � an � am � n, (am)n � amn,and am an � am � n.
Property of Equality for If b is a positive number other than 1,Exponential Functions then bx � by if and only if x � y.
Property of Inequality forIf b � 1
Exponential Functionsthen bx � by if and only if x � yand bx � by if and only if x � y.
Study Guide and Intervention (continued)
Exponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Solve 4x � 1 � 2x � 5.4x � 1 � 2x � 5 Original equation
(22)x � 1 � 2x � 5 Rewrite 4 as 22.
2(x � 1) � x � 5 Prop. of Inequality for ExponentialFunctions
2x � 2 � x � 5 Distributive Property
x � 7 Subtract x and add 2 to each side.
Solve 52x � 1 � .
52x � 1 � Original inequality
52x � 1 � 5�3 Rewrite as 5�3.
2x � 1 � �3 Prop. of Inequality for Exponential Functions
2x � �2 Add 1 to each side.
x � �1 Divide each side by 2.
The solution set is {x|x � �1}.
1�125
1�125
1�125
Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify each expression.
1. (3�2�)�2� 2. 25�2� � 125�2� 3. (x�2�y3�2�)�2�
9 55�2� or 3125�2� x2y6
4. (x�6�)(x�5�) 5. (x�6�)�5� 6. (2x)(5x3)x�6� � �5� x�30� 10x4�
Solve each equation or inequality. Check your solution.
7. 32x � 1 � 3x � 2 3 8. 23x � 4x � 2 4 9. 32x � 1 � �
10. 4x � 1 � 82x � 3 � 11. 8x � 2 � 12. 252x � 125x � 2 6
13. 4�x� � 16�5� 20 14. x�3� � 36���34�
6 15. x�2� � 81��18�
�
3
16. 3x � 4 � x � 1 17. 42x � 2 � 2x � 1 x � 18. 52x � 125x � 5 x � 15
19. 104x � 1 � 100x � 2 20. 73x � 49x2 21. 82x � 5 � 4x � 8
x � � x � or x � 0 x � �341�
3�2
5�2
5�3
1�27
2�3
1�16
7�4
1�2
1�9
Skills PracticeExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 575 Glencoe Algebra 2
Less
on
10-
1
Sketch the graph of each function. Then state the function’s domain and range.
1. y � 3(2)x 2. y � 2� �x
domain: all real numbers; domain: all real numbers;range: all positive numbers range: all positive numbers
Determine whether each function represents exponential growth or decay.
3. y � 3(6)x growth 4. y � 2� �xdecay
5. y � 10�x decay 6. y � 2(2.5)x growth
Write an exponential function whose graph passes through the given points.
7. (0, 1) and (�1, 3) y � � �x8. (0, 4) and (1, 12) y � 4(3)x
9. (0, 3) and (�1, 6) y � 3� �x10. (0, 5) and (1, 15) y � 5(3)x
11. (0, 0.1) and (1, 0.5) y � 0.1(5)x 12. (0, 0.2) and (1, 1.6) y � 0.2(8)x
Simplify each expression.
13. (3�3�)�3� 27 14. (x�2�)�7� x�14�
15. 52�3� � 54�3� 56�3� 16. x3 x x2�
Solve each equation or inequality. Check your solution.
17. 3x � 9 x � 2 18. 22x � 3 � 32 1
19. 49x � x � � 20. 43x � 2 � 16
21. 32x � 5 � 27x 5 22. 27x � 32x � 3 3
4�3
1�2
1�7
1�2
1�3
9�10
x
y
Ox
y
O
1�2
© Glencoe/McGraw-Hill 576 Glencoe Algebra 2
Sketch the graph of each function. Then state the function’s domain and range.
1. y � 1.5(2)x 2. y � 4(3)x 3. y � 3(0.5)x
domain: all real domain: all real domain: all real numbers; range: all numbers; range: all numbers; range: all positive numbers positive numbers positive numbers
Determine whether each function represents exponential growth or decay.
4. y � 5(0.6)x decay 5. y � 0.1(2)x growth 6. y � 5 � 4�x decay
Write an exponential function whose graph passes through the given points.
7. (0, 1) and (�1, 4) 8. (0, 2) and (1, 10) 9. (0, �3) and (1, �1.5)
y � � �xy � 2(5)x y � �3(0.5)x
10. (0, 0.8) and (1, 1.6) 11. (0, �0.4) and (2, �10) 12. (0, ) and (3, 8)
y � 0.8(2)x y � �0.4(5)x y � �(2)x
Simplify each expression.
13. (2�2�)�8� 16 14. (n�3�)�75� n15 15. y�6� � y5�6� y6�6�
16. 13�6� � 13�24� 133�6� 17. n3 n n3 � � 18. 125�11� 5�11� 52�11�
Solve each equation or inequality. Check your solution.
19. 33x � 5 � 81 x � 3 20. 76x � 72x � 20 �5 21. 36n � 5 � 94n � 3 n �
22. 92x � 1 � 27x � 4 14 23. 23n � 1 � � �nn 24. 164n � 1 � 1282n � 1
BIOLOGY For Exercises 25 and 26, use the following information.The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.
25. Write an exponential function to model the population y of bacteria after x days.y � 12,000(2)x
26. How many bacteria are there after 6 days? 768,000
27. EDUCATION A college with a graduating class of 4000 students in the year 2002predicts that it will have a graduating class of 4862 in 4 years. Write an exponentialfunction to model the number of students y in the graduating class t years after 2002.y � 4000(1.05)t
11�2
1�6
1�8
1�2
1�4
x
y
Ox
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Practice (Average)
Exponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Reading to Learn MathematicsExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 577 Glencoe Algebra 2
Less
on
10-
1
Pre-Activity How does an exponential function describe tournament play?
Read the introduction to Lesson 10-1 at the top of page 523 in your textbook.
How many rounds of play would be needed for a tournament with 100players? 7
Reading the Lesson
1. Indicate whether each of the following statements about the exponential function y � 10x is true or false.
a. The domain is the set of all positive real numbers. false
b. The y-intercept is 1. true
c. The function is one-to-one. true
d. The y-axis is an asymptote of the graph. false
e. The range is the set of all real numbers. false
2. Determine whether each function represents exponential growth or decay.
a. y � 0.2(3)x. growth b. y � 3� �x. decay c. y � 0.4(1.01)x. growth
3. Supply the reason for each step in the following solution of an exponential equation.
92x � 1 � 27x Original equation
(32)2x � 1 � (33)x Rewrite each side with a base of 3.32(2x � 1) � 33x Power of a Power
2(2x � 1) � 3x Property of Equality for Exponential Functions4x � 2 � 3x Distributive Propertyx � 2 � 0 Subtract 3x from each side.
x � 2 Add 2 to each side.
Helping You Remember
4. One way to remember that polynomial functions and exponential functions are differentis to contrast the polynomial function y � x2 and the exponential function y � 2x. Tell atleast three ways they are different.
Sample answer: In y � x2, the variable x is a base, but in y � 2x, thevariable x is an exponent. The graph of y � x2 is symmetric with respectto the y-axis, but the graph of y � 2x is not. The graph of y � x2 touchesthe x-axis at (0, 0), but the graph of y � 2x has the x-axis as an asymptote.You can compute the value of y � x2 mentally for x � 100, but you cannotcompute the value of y � 2x mentally for x � 100.
2�5
© Glencoe/McGraw-Hill 578 Glencoe Algebra 2
Finding Solutions of xy � yx
Perhaps you have noticed that if x and y are interchanged in equations suchas x � y and xy � 1, the resulting equation is equivalent to the originalequation. The same is true of the equation xy � yx. However, findingsolutions of xy � yx and drawing its graph is not a simple process.
Solve each problem. Assume that x and y are positive real numbers.
1. If a � 0, will (a, a) be a solution of xy � yx? Justify your answer.
2. If c � 0, d � 0, and (c, d) is a solution of xy � yx, will (d, c) also be a solution? Justify your answer.
3. Use 2 as a value for y in xy � yx. The equation becomes x2 � 2x.
a. Find equations for two functions, f(x) and g(x) that you could graph tofind the solutions of x2 � 2x. Then graph the functions on a separatesheet of graph paper.
b. Use the graph you drew for part a to state two solutions for x2 � 2x.Then use these solutions to state two solutions for xy � yx.
4. In this exercise, a graphing calculator will be very helpful. Use the technique of Exercise 3 to complete the tables below. Then graph xy � yx
for positive values of x and y. If there are asymptotes, show them in yourdiagram using dotted lines. Note that in the table, some values of y callfor one value of x, others call for two.
x
y
O
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4
4
5
5
8
8
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
x y
�12
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�34
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1
2
2
3
3
Study Guide and InterventionLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 579 Glencoe Algebra 2
Less
on
10-
2
Logarithmic Functions and Expressions
Definition of Logarithm Let b and x be positive numbers, b � 1. The logarithm of x with base b is denoted with Base b logb x and is defined as the exponent y that makes the equation by � x true.
The inverse of the exponential function y � bx is the logarithmic function x � by.This function is usually written as y � logb x.
1. The function is continuous and one-to-one.
Properties of2. The domain is the set of all positive real numbers.
Logarithmic Functions3. The y-axis is an asymptote of the graph.4. The range is the set of all real numbers.5. The graph contains the point (0, 1).
Write an exponential equation equivalent to log3 243 � 5.35 � 243
Write a logarithmic equation equivalent to 6�3 � .
log6 � �3
Evaluate log8 16.
8�43
�
� 16, so log8 16 � .
Write each equation in logarithmic form.
1. 27 � 128 2. 3�4 � 3. � �3�
log2 128 � 7 log3 � �4 log�17
� � 3
Write each equation in exponential form.
4. log15 225 � 2 5. log3 � �3 6. log4 32 �
152 � 225 3�3 � 4�52
�� 32
Evaluate each expression.
7. log4 64 3 8. log2 64 6 9. log100 100,000 2.5
10. log5 625 4 11. log27 81 12. log25 5
13. log2 �7 14. log10 0.00001 �5 15. log4 �2.51�32
1�128
1�2
4�3
1�27
5�2
1�27
1�343
1�81
1�343
1�7
1�81
4�3
1�216
1�216
Example 1Example 1
Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 580 Glencoe Algebra 2
Solve Logarithmic Equations and Inequalities
Logarithmic to If b � 1, x � 0, and logb x � y, then x � by.Exponential Inequality If b � 1, x � 0, and logb x � y, then 0 � x � by.
Property of Equality for If b is a positive number other than 1, Logarithmic Functions then logb x � logb y if and only if x � y.
Property of Inequality for If b � 1, then logb x � logb y if and only if x � y, Logarithmic Functions and logb x � logb y if and only if x � y.
Study Guide and Intervention (continued)
Logarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Solve log2 2x � 3.log2 2x � 3 Original equation
2x � 23 Definition of logarithm
2x � 8 Simplify.
x � 4 Simplify.
The solution is x � 4.
Solve log5 (4x � 3) � 3.log5 (4x � 3) � 3 Original equation
0 � 4x � 3 � 53 Logarithmic to exponential inequality
3 � 4x � 125 � 3 Addition Property of Inequalities
� x � 32 Simplify.
The solution set is �x | � x � 32�.3�4
3�4
Example 1Example 1 Example 2Example 2
ExercisesExercises
Solve each equation or inequality.
1. log2 32 � 3x 2. log3 2c � �2
3. log2x 16 � �2 4. log25 � � � 10
5. log4 (5x � 1) � 2 3 6. log8 (x � 5) � 9
7. log4 (3x � 1) � log4 (2x � 3) 4 8. log2 (x2 � 6) � log2 (2x � 2) 4
9. logx � 4 27 � 3 �1 10. log2 (x �3) � 4 13
11. logx 1000 � 3 10 12. log8 (4x � 4) � 2 15
13. log2 2x � 2 x � 2 14. log5 x � 2 x � 25
15. log2 (3x � 1) � 4 � � x � 5 16. log4 (2x) � � x �
17. log3 (x � 3) � 3 �3 � x � 24 18. log27 6x � x �3�2
2�3
1�4
1�2
1�3
2�3
1�2
x�2
1�8
1�18
5�3
Skills PracticeLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 581 Glencoe Algebra 2
Less
on
10-
2
Write each equation in logarithmic form.
1. 23 � 8 log2 8 � 3 2. 32 � 9 log3 9 � 2
3. 8�2 � log8 � �2 4. � �2� log�
13
� � 2
Write each equation in exponential form.
5. log3 243 � 5 35 � 243 6. log4 64 � 3 43 � 64
7. log9 3 � 9�12
�� 3 8. log5 � �2 5�2 �
Evaluate each expression.
9. log5 25 2 10. log9 3
11. log10 1000 3 12. log125 5
13. log4 �3 14. log5 �4
15. log8 83 3 16. log27 �
Solve each equation or inequality. Check your solutions.
17. log3 x � 5 243 18. log2 x � 3 8
19. log4 y � 0 0 � y � 1 20. log�14
� x � 3
21. log2 n � �2 n � 22. logb 3 � 9
23. log6 (4x � 12) � 2 6 24. log2 (4x � 4) � 5 x � 9
25. log3 (x � 2) � log3 (3x) 1 26. log6 (3y � 5) � log6 (2y � 3) y 8
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1�4
1�64
1�3
1�3
1�625
1�64
1�3
1�2
1�25
1�25
1�2
1�9
1�9
1�3
1�64
1�64
© Glencoe/McGraw-Hill 582 Glencoe Algebra 2
Write each equation in logarithmic form.
1. 53 � 125 log5 125 � 3 2. 70 � 1 log7 1 � 0 3. 34 � 81 log3 81 � 4
4. 3�4 � 5. � �3� 6. 7776
�15
�
� 6
log3 � �4 log�14
� � 3 log7776 6 �
Write each equation in exponential form.
7. log6 216 � 3 63 � 216 8. log2 64 � 6 26 � 64 9. log3 � �4 3�4 �
10. log10 0.00001 � �5 11. log25 5 � 12. log32 8 �
10�5 � 0.00001 25�12
�� 5 32
�35
�� 8
Evaluate each expression.
13. log3 81 4 14. log10 0.0001 �4 15. log2 �4 16. log�13
� 27 �3
17. log9 1 0 18. log8 4 19. log7 �2 20. log6 64 4
21. log3 �1 22. log4 �4 23. log9 9(n � 1) n � 1 24. 2log2 32 32
Solve each equation or inequality. Check your solutions.
25. log10 n � �3 26. log4 x � 3 x � 64 27. log4 x � 8
28. log�15
� x � �3 125 29. log7 q � 0 0 � q � 1 30. log6 (2y � 8) � 2 y 14
31. logy 16 � �4 32. logn � �3 2 33. logb 1024 � 5 4
34. log8 (3x � 7) � log8 (7x � 4) 35. log7 (8x � 20) � log7 (x � 6) 36. log3 (x2 � 2) � log3 x
x � �2 2
37. SOUND Sounds that reach levels of 130 decibels or more are painful to humans. Whatis the relative intensity of 130 decibels? 1013
38. INVESTING Maria invests $1000 in a savings account that pays 8% interestcompounded annually. The value of the account A at the end of five years can bedetermined from the equation log A � log[1000(1 � 0.08)5]. Find the value of A to thenearest dollar. $1469
3�4
1�8
1�2
3�2
1�1000
1�256
1�3
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1�2
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1�81
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1�64
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Practice (Average)
Logarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Reading to Learn MathematicsLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 583 Glencoe Algebra 2
Less
on
10-
2
Pre-Activity Why is a logarithmic scale used to measure sound?
Read the introduction to Lesson 10-2 at the top of page 531 in your textbook.
How many times louder than a whisper is normal conversation?104 or 10,000 times
Reading the Lesson1. a. Write an exponential equation that is equivalent to log3 81 � 4. 34 � 81
b. Write a logarithmic equation that is equivalent to 25��12
�� . log25 � �
c. Write an exponential equation that is equivalent to log4 1 � 0. 40 � 1
d. Write a logarithmic equation that is equivalent to 10�3 � 0.001. log10 0.001 � �3
e. What is the inverse of the function y � 5x? y � log5 x
f. What is the inverse of the function y � log10 x? y � 10x
2. Match each function with its graph.
a. y � 3x IV b. y � log3 x I c. y � � �xII
I. II. III.
3. Indicate whether each of the following statements about the exponential function y � log5 x is true or false.
a. The y-axis is an asymptote of the graph. trueb. The domain is the set of all real numbers. falsec. The graph contains the point (5, 0). falsed. The range is the set of all real numbers. truee. The y-intercept is 1. false
Helping You Remember4. An important skill needed for working with logarithms is changing an equation between
logarithmic and exponential forms. Using the words base, exponent, and logarithm, describean easy way to remember and apply the part of the definition of logarithm that says,“logb x � y if and only if by � x.” Sample answer: In these equations, b standsfor base. In log form, b is the subscript, and in exponential form, b is thenumber that is raised to a power. A logarithm is an exponent, so y, which isthe log in the first equation, becomes the exponent in the second equation.
x
y
Ox
y
O
x
y
O
1�3
1�2
1�5
1�5
© Glencoe/McGraw-Hill 584 Glencoe Algebra 2
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-210-2
Musical RelationshipsThe frequencies of notes in a musical scale that are one octave apart arerelated by an exponential equation. For the eight C notes on a piano, theequation is Cn � C12n � 1, where Cn represents the frequency of note Cn.
1. Find the relationship between C1 and C2.
2. Find the relationship between C1 and C4.
The frequencies of consecutive notes are related by a common ratio r. The general equation is fn � f1rn � 1.
3. If the frequency of middle C is 261.6 cycles per second and the frequency of the next higher C is 523.2 cycles per second, find the common ratio r. (Hint: The two C’s are 12 notes apart.) Write the answer as a radicalexpression.
4. Substitute decimal values for r and f1 to find a specific equation for fn.
5. Find the frequency of F# above middle C.
6. Frets are a series of ridges placed across the fingerboard of a guitar. Theyare spaced so that the sound made by pressing a string against one frethas about 1.0595 times the wavelength of the sound made by using thenext fret. The general equation is wn � w0(1.0595)n. Describe thearrangement of the frets on a guitar.
Study Guide and InterventionProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 585 Glencoe Algebra 2
Less
on
10-
3
Properties of Logarithms Properties of exponents can be used to develop thefollowing properties of logarithms.
Product Property For all positive numbers m, n, and b, where b � 1, of Logarithms logb mn � logb m � logb n.
Quotient Property For all positive numbers m, n, and b, where b � 1, of Logarithms logb �
mn
� � logb m � logb n.
Power Property For any real number p and positive numbers m and b, of Logarithms where b � 1, logb mp � p logb m.
Use log3 28 � 3.0331 and log3 4 � 1.2619 to approximate the value of each expression.
ExampleExample
a. log3 36
log3 36 � log3 (32 � 4)� log3 32 � log3 4� 2 � log3 4� 2 � 1.2619� 3.2619
b. log3 7
log3 7 � log3 � �� log3 28 � log3 4� 3.0331 � 1.2619� 1.7712
c. log3 256
log3 256 � log3 (44)� 4 � log3 4� 4(1.2619)� 5.0476
28�4
ExercisesExercises
Use log12 3 � 0.4421 and log12 7 � 0.7831 to evaluate each expression.
1. log12 21 1.2252 2. log12 0.3410 3. log12 49 1.5662
4. log12 36 1.4421 5. log12 63 1.6673 6. log12 �0.2399
7. log12 0.2022 8. log12 16,807 3.9155 9. log12 441 2.4504
Use log5 3 � 0.6826 and log5 4 � 0.8614 to evaluate each expression.
10. log5 12 1.5440 11. log5 100 2.8614 12. log5 0.75 �0.1788
13. log5 144 3.0880 14. log5 0.3250 15. log5 375 3.6826
16. log5 1.3� 0.1788 17. log5 �0.3576 18. log5 1.730481�5
9�16
27�16
81�49
27�49
7�3
© Glencoe/McGraw-Hill 586 Glencoe Algebra 2
Solve Logarithmic Equations You can use the properties of logarithms to solveequations involving logarithms.
Solve each equation.
a. 2 log3 x � log3 4 � log3 25
2 log3 x � log3 4 � log3 25 Original equation
log3 x2 � log3 4 � log3 25 Power Property
log3 � log3 25 Quotient Property
� 25 Property of Equality for Logarithmic Functions
x2 � 100 Multiply each side by 4.
x � �10 Take the square root of each side.
Since logarithms are undefined for x � 0, �10 is an extraneous solution.The only solution is 10.
b. log2 x � log2 (x � 2) � 3
log2 x � log2 (x � 2) � 3 Original equation
log2 x(x � 2) � 3 Product Property
x(x � 2) � 23 Definition of logarithm
x2 � 2x � 8 Distributive Property
x2 � 2x � 8 � 0 Subtract 8 from each side.
(x � 4)(x � 2) � 0 Factor.
x � 2 or x � �4 Zero Product Property
Since logarithms are undefined for x � 0, �4 is an extraneous solution.The only solution is 2.
Solve each equation. Check your solutions.
1. log5 4 � log5 2x � log5 24 3 2. 3 log4 6 � log4 8 � log4 x 27
3. log6 25 � log6 x � log6 20 4 4. log2 4 � log2 (x � 3) � log2 8 �
5. log6 2x � log6 3 � log6 (x � 1) 3 6. 2 log4 (x � 1) � log4 (11 � x) 2
7. log2 x � 3 log2 5 � 2 log2 10 12,500 8. 3 log2 x � 2 log2 5x � 2 100
9. log3 (c � 3) � log3 (4c � 1) � log3 5 10. log5 (x � 3) � log5 (2x � 1) � 24�7
8�19
5�2
1�2
x2�4
x2�4
Study Guide and Intervention (continued)
Properties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
ExampleExample
ExercisesExercises
Skills PracticeProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 587 Glencoe Algebra 2
Less
on
10-
3
Use log2 3 � 1.5850 and log2 5 � 2.3219 to approximate the value of eachexpression.
1. log2 25 4.6438 2. log2 27 4.755
3. log2 �0.7369 4. log2 0.7369
5. log2 15 3.9069 6. log2 45 5.4919
7. log2 75 6.2288 8. log2 0.6 �0.7369
9. log2 �1.5850 10. log2 0.8481
Solve each equation. Check your solutions.
11. log10 27 � 3 log10 x 3 12. 3 log7 4 � 2 log7 b 8
13. log4 5 � log4 x � log4 60 12 14. log6 2c � log6 8 � log6 80 5
15. log5 y � log5 8 � log5 1 8 16. log2 q � log2 3 � log2 7 21
17. log9 4 � 2 log9 5 � log9 w 100 18. 3 log8 2 � log8 4 � log8 b 2
19. log10 x � log10 (3x � 5) � log10 2 2 20. log4 x � log4 (2x � 3) � log4 2 2
21. log3 d � log3 3 � 3 9 22. log10 y � log10 (2 � y) � 0 1
23. log2 s � 2 log2 5 � 0 24. log2 (x � 4) � log2 (x � 3) � 3 4
25. log4 (n � 1) � log4 (n � 2) � 1 3 26. log5 10 � log5 12 � 3 log5 2 � log5 a 15
1�25
9�5
1�3
5�3
3�5
© Glencoe/McGraw-Hill 588 Glencoe Algebra 2
Use log10 5 � 0.6990 and log10 7 � 0.8451 to approximate the value of eachexpression.
1. log10 35 1.5441 2. log10 25 1.3980 3. log10 0.1461 4. log10 �0.1461
5. log10 245 2.3892 6. log10 175 2.2431 7. log10 0.2 �0.6990 8. log10 0.5529
Solve each equation. Check your solutions.
9. log7 n � log7 8 4 10. log10 u � log10 4 8
11. log6 x � log6 9 � log6 54 6 12. log8 48 � log8 w � log8 4 12
13. log9 (3u � 14) � log9 5 � log9 2u 2 14. 4 log2 x � log2 5 � log2 405 3
15. log3 y � �log3 16 � log3 64 16. log2 d � 5 log2 2 � log2 8 4
17. log10 (3m � 5) � log10 m � log10 2 2 18. log10 (b � 3) � log10 b � log10 4 1
19. log8 (t � 10) � log8 (t � 1) � log8 12 2 20. log3 (a � 3) � log3 (a � 2) � log3 6 0
21. log10 (r � 4) � log10 r � log10 (r � 1) 2 22. log4 (x2 � 4) � log4 (x � 2) � log4 1 3
23. log10 4 � log10 w � 2 25 24. log8 (n � 3) � log8 (n � 4) � 1 4
25. 3 log5 (x2 � 9) � 6 � 0 �4 26. log16 (9x � 5) � log16 (x2 � 1) � 3
27. log6 (2x � 5) � 1 � log6 (7x � 10) 8 28. log2 (5y � 2) � 1 � log2 (1 � 2y) 0
29. log10 (c2 � 1) � 2 � log10 (c � 1) 101 30. log7 x � 2 log7 x � log7 3 � log7 72 6
31. SOUND The loudness L of a sound in decibels is given by L � 10 log10 R, where R is thesound’s relative intensity. If the intensity of a certain sound is tripled, by how manydecibels does the sound increase? about 4.8 db
32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitudereading m is given by m � log10 x, where x represents the amplitude of the seismic wavecausing ground motion. How many times greater is the amplitude of an earthquake thatmeasures 4.5 on the Richter scale than one that measures 3.5? 10 times
1�2
1�4
1�3
3�2
2�3
25�7
5�7
7�5
Practice (Average)
Properties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
Reading to Learn MathematicsProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 589 Glencoe Algebra 2
Less
on
10-
3
Pre-Activity How are the properties of exponents and logarithms related?
Read the introduction to Lesson 10-3 at the top of page 541 in your textbook.
Find the value of log5 125. 3 Find the value of log5 5. 1Find the value of log5 (125 � 5). 2Which of the following statements is true? BA. log5 (125 � 5) � (log5 125) � (log5 5)
B. log5 (125 � 5) � log5 125 � log5 5
Reading the Lesson1. Each of the properties of logarithms can be stated in words or in symbols. Complete the
statements of these properties in words.
a. The logarithm of a quotient is the of the logarithms of the
and the .
b. The logarithm of a power is the of the logarithm of the base and
the .
c. The logarithm of a product is the of the logarithms of its
.
2. State whether each of the following equations is true or false. If the statement is true,name the property of logarithms that is illustrated.
a. log3 10 � log3 30 � log3 3 true; Quotient Propertyb. log4 12 � log4 4 � log4 8 falsec. log2 81 � 2 log2 9 true; Power Propertyd. log8 30 � log8 5 � log8 6 false
3. The algebraic process of solving the equation log2 x � log2 (x � 2) � 3 leads to “x � �4or x � 2.” Does this mean that both �4 and 2 are solutions of the logarithmic equation?Explain your reasoning. Sample answer: No; 2 is a solution because it checks: log2 2 � log2 (2 � 2) � log2 2 � log2 4 � 1 � 2 � 3. However,because log2 (�4) and log2 (� 2) are undefined, �4 is an extraneoussolution and must be eliminated. The only solution is 2.
Helping You Remember4. A good way to remember something is to relate it something you already know. Use words
to explain how the Product Property for exponents can help you remember the productproperty for logarithms. Sample answer: When you multiply two numbers orexpressions with the same base, you add the exponents and keep thesame base. Logarithms are exponents, so to find the logarithm of aproduct, you add the logarithms of the factors, keeping the same base.
factorssum
exponentproduct
denominatornumeratordifference
© Glencoe/McGraw-Hill 590 Glencoe Algebra 2
SpiralsConsider an angle in standard position with its vertex at a point O called thepole. Its initial side is on a coordinatized axis called the polar axis. A point Pon the terminal side of the angle is named by the polar coordinates (r, �),where r is the directed distance of the point from O and � is the measure ofthe angle. Graphs in this system may be drawn on polar coordinate papersuch as the kind shown below.
1. Use a calculator to complete the table for log2r � �12�0�.
(Hint: To find � on a calculator, press 120 r 2 .)
2. Plot the points found in Exercise 1 on the grid above and connect to form a smooth curve.
This type of spiral is called a logarithmic spiral because the angle measures are proportional to the logarithms of the radii.
r 1 2 3 4 5 6 7 8
) LOG�) LOG�
0
10
20
30
40
5060
708090100
110120
130
140
150
160
170
180
190
200
210
220
230
240250
260 270 280290
300310
320
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350
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
Study Guide and InterventionCommon Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 591 Glencoe Algebra 2
Less
on
10-
4
Common Logarithms Base 10 logarithms are called common logarithms. Theexpression log10 x is usually written without the subscript as log x. Use the key onyour calculator to evaluate common logarithms.The relation between exponents and logarithms gives the following identity.
Inverse Property of Logarithms and Exponents 10log x � x
Evaluate log 50 to four decimal places.Use the LOG key on your calculator. To four decimal places, log 50 � 1.6990.
Solve 32x � 1 � 12.32x � 1 � 12 Original equation
log 32x � 1 � log 12 Property of Equality for Logarithms
(2x � 1) log 3 � log 12 Power Property of Logarithms
2x � 1 � Divide each side by log 3.
2x � � 1 Subtract 1 from each side.
x � � � 1� Multiply each side by .
x � 0.6309
Use a calculator to evaluate each expression to four decimal places.
1. log 18 2. log 39 3. log 1201.2553 1.5911 2.0792
4. log 5.8 5. log 42.3 6. log 0.0030.7634 1.6263 �2.5229
Solve each equation or inequality. Round to four decimal places.
7. 43x � 12 0.5975 8. 6x � 2 � 18 �0.3869
9. 54x � 2 � 120 1.2437 10. 73x � 1 � 21 {x |x � 0.8549}
11. 2.4x � 4 � 30 �0.1150 12. 6.52x � 200 {x |x � 1.4153}
13. 3.64x � 1 � 85.4 1.1180 14. 2x � 5 � 3x � 2 13.9666
15. 93x � 45x � 2 �8.1595 16. 6x � 5 � 27x � 3 �3.6069
1�2
log 12�log 3
1�2
log 12�log 3
log 12�log 3
LOG
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 592 Glencoe Algebra 2
Change of Base Formula The following formula is used to change expressions withdifferent logarithmic bases to common logarithm expressions.
Change of Base Formula For all positive numbers a, b, and n, where a � 1 and b � 1, loga n �
Express log8 15 in terms of common logarithms. Then approximateits value to four decimal places.
log8 15 � Change of Base Formula
� 1.3023 Simplify.
The value of log8 15 is approximately 1.3023.
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
1. log3 16 2. log2 40 3. log5 35
, 2.5237 , 5.3219 , 2.2091
4. log4 22 5. log12 200 6. log2 50
, 2.2297 , 2.1322 , 5.6439
7. log5 0.4 8. log3 2 9. log4 28.5
, �0.5693 , 0.6309 , 2.4164
10. log3 (20)2 11. log6 (5)4 12. log8 (4)5
, 5.4537 , 3.5930 , 3.3333
13. log5 (8)3 14. log2 (3.6)6 15. log12 (10.5)4
, 3.8761 , 11.0880 , 3.7851
16. log3 �150� 17. log43�39� 18. log5
4�1600�
, 2.2804 , 0.8809 , 1.1460log 1600��4 log 5
log 39�3 log 4
log 150�2 log 3
4 log 10.5��
log 126 log 3.6��
log 23 log 8�log 5
5 log 4�log 8
4 log 5�log 6
2 log 20��
log 3
log 28.5��
log 4log 2�log 3
log 0.4�log 5
log 50�log 2
log 200�log 12
log 22�log 4
log 35�log 5
log 40�log 2
log 16�log 3
log10 15�log10 8
logb n�logb a
Study Guide and Intervention (continued)
Common Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
ExampleExample
ExercisesExercises
Skills PracticeCommon Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 593 Glencoe Algebra 2
Less
on
10-
4
Use a calculator to evaluate each expression to four decimal places.
1. log 6 0.7782 2. log 15 1.1761
3. log 1.1 0.0414 4. log 0.3 �0.5229
Use the formula pH � �log[H�] to find the pH of each substance given itsconcentration of hydrogen ions.
5. gastric juices: [H�] � 1.0 � 10�1 mole per liter 1.0
6. tomato juice: [H�] � 7.94 � 10�5 mole per liter 4.1
7. blood: [H�] � 3.98 � 10�8 mole per liter 7.4
8. toothpaste: [H�] � 1.26 � 10�10 mole per liter 9.9
Solve each equation or inequality. Round to four decimal places.
9. 3x � 243 {x |x � 5} 10. 16v �v �v � � �11. 8p � 50 1.8813 12. 7y � 15 1.3917
13. 53b � 106 0.9659 14. 45k � 37 0.5209
15. 127p � 120 0.2752 16. 92m � 27 0.75
17. 3r � 5 � 4.1 6.2843 18. 8y � 4 � 15 {y |y � �2.6977}
19. 7.6d � 3 � 57.2 �1.0048 20. 0.5t � 8 � 16.3 3.9732
21. 42x2� 84 �1.0888 22. 5x2 � 1� 10 �0.6563
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
23. log3 7 ; 1.7712 24. log5 66 ; 2.6032
25. log2 35 ; 5.1293 26. log6 10 ; 1.2851log10 10��log10 6
log10 35��log10 2
log10 66��log10 5
log10 7�log10 3
1�2
1�4
© Glencoe/McGraw-Hill 594 Glencoe Algebra 2
Use a calculator to evaluate each expression to four decimal places.
1. log 101 2.0043 2. log 2.2 0.3424 3. log 0.05 �1.3010
Use the formula pH � �log[H�] to find the pH of each substance given itsconcentration of hydrogen ions.
4. milk: [H�] � 2.51 � 10�7 mole per liter 6.6
5. acid rain: [H�] � 2.51 � 10�6 mole per liter 5.6
6. black coffee: [H�] � 1.0 � 10�5 mole per liter 5.0
7. milk of magnesia: [H�] � 3.16 � 10�11 mole per liter 10.5
Solve each equation or inequality. Round to four decimal places.
8. 2x 25 {x |x 4.6439} 9. 5a � 120 2.9746 10. 6z � 45.6 2.1319
11. 9m � 100 {m |m � 2.0959} 12. 3.5x � 47.9 3.0885 13. 8.2y � 64.5 1.9802
14. 2b � 1 7.31 {b |b � 1.8699} 15. 42x � 27 1.1887 16. 2a � 4 � 82.1 10.3593
17. 9z � 2 � 38 {z |z � 3.6555} 18. 5w � 3 � 17 �1.2396 19. 30x2� 50 �1.0725
20. 5x2 � 3 � 72 �2.3785 21. 42x � 9x � 1 3.8188 22. 2n � 1 � 52n � 1 0.9117
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
23. log5 12 ; 1.5440 24. log8 32 ; 1.6667 25. log11 9 ; 0.9163
26. log2 18 ; 4.1699 27. log9 6 ; 0.8155 28. log7 �8� ;
29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H�]in the soil is not less than 1.58 � 10�8 mole per liter. What is the pH of the soil in whichthese irises will flourish? 7.8 or less
30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater isthe hydrogen ion concentration of vinegar than of milk? about 5000
31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubleseach hour. The number of bacteria N present after t hours is N � 1000(2) t. How long willit take the culture to increase to 50,000 bacteria? about 5.6 h
32. SOUND An equation for loudness L in decibels is given by L � 10 log R, where R is thesound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noisecan reach 120 decibels. How many times greater is the relative intensity of the air-raidsiren than that of the jet engine noise? 1000
log10 8�2 log10 7
log10 6��log10 9
log10 18��log10 2
log10 9��log10 11
log10 32��log10 8
log10 12��log10 5
Practice (Average)
Common Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
0.5343
Reading to Learn MathematicsCommon Logarithms
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 595 Glencoe Algebra 2
Less
on
10-
4
Pre-Activity Why is a logarithmic scale used to measure acidity?
Read the introduction to Lesson 10-4 at the top of page 547 in your textbook.
Which substance is more acidic, milk or tomatoes? tomatoes
Reading the Lesson
1. Rhonda used the following keystrokes to enter an expression on her graphing calculator:
17
The calculator returned the result 1.230448921.Which of the following conclusions are correct? a, c, and d
a. The base 10 logarithm of 17 is about 1.2304.
b. The base 17 logarithm of 10 is about 1.2304.
c. The common logarithm of 17 is about 1.230449.
d. 101.230448921 is very close to 17.
e. The common logarithm of 17 is exactly 1.230448921.
2. Match each expression from the first column with an expression from the second columnthat has the same value.
a. log2 2 iv i. log4 1
b. log 12 iii ii. log2 8
c. log3 1 i iii. log10 12
d. log5 v iv. log5 5
e. log 1000 ii v. log 0.1
3. Calculators do not have keys for finding base 8 logarithms directly. However, you can use
a calculator to find log8 20 if you apply the formula.
Which of the following expressions are equal to log8 20? B and C
A. log20 8 B. C. D.
Helping You Remember
4. Sometimes it is easier to remember a formula if you can state it in words. State thechange of base formula in words. Sample answer: To change the logarithm of anumber from one base to another, divide the log of the original numberin the old base by the log of the new base in the old base.
log 8�log 20
log 20�log 8
log10 20�log10 8
change of base
1�5
ENTER) LOG
© Glencoe/McGraw-Hill 596 Glencoe Algebra 2
The Slide RuleBefore the invention of electronic calculators, computations were oftenperformed on a slide rule. A slide rule is based on the idea of logarithms. It hastwo movable rods labeled with C and D scales. Each of the scales is logarithmic.
To multiply 2 � 3 on a slide rule, move the C rod to the right as shownbelow. You can find 2 � 3 by adding log 2 to log 3, and the slide rule adds thelengths for you. The distance you get is 0.778, or the logarithm of 6.
Follow the steps to make a slide rule.
1. Use graph paper that has small squares, such as 10 squares to the inch. Using the scales shown at the right, plot the curve y � log x for x � 1, 1.5,and the whole numbers from 2 through 10. Make an obvious heavy dot for each point plotted.
2. You will need two strips of cardboard. A 5-by-7 index card, cut in half the long way,will work fine. Turn the graph you made in Exercise 1 sideways and use it to marka logarithmic scale on each of the twostrips. The figure shows the mark for 2 being drawn.
3. Explain how to use a slide rule to divide 8 by 2.
0
0.1
0.2
0.3 y
12
1 1.5 2
y = log x
0.1
0.2
1 2
1
21
CD
2
4
3
6
4 5 6 7 8 9
83 5 7 9
log 6
log 3log 2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
C
D
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
Study Guide and InterventionBase e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 597 Glencoe Algebra 2
Less
on
10-
5
Base e and Natural Logarithms The irrational number e � 2.71828… often occursas the base for exponential and logarithmic functions that describe real-world phenomena.
Natural Base e As n increases, �1 � �napproaches e � 2.71828….
ln x � loge x
The functions y � ex and y � ln x are inverse functions.
Inverse Property of Base e and Natural Logarithms eln x � x ln ex � x
Natural base expressions can be evaluated using the ex and ln keys on your calculator.
Evaluate ln 1685.Use a calculator.ln 1685 � 7.4295
Write a logarithmic equation equivalent to e2x � 7.e2x � 7 → loge 7 � 2x or 2x � ln 7
Evaluate ln e18.Use the Inverse Property of Base e and Natural Logarithms.ln e18 � 18
Use a calculator to evaluate each expression to four decimal places.
1. ln 732 2. ln 84,350 3. ln 0.735 4. ln 1006.5958 11.3427 �0.3079 4.6052
5. ln 0.0824 6. ln 2.388 7. ln 128,245 8. ln 0.00614�2.4962 0.8705 11.7617 �5.0929
Write an equivalent exponential or logarithmic equation.
9. e15 � x 10. e3x � 45 11. ln 20 � x 12. ln x � 8ln x � 15 3x � ln 45 ex � 20 x � e8
13. e�5x � 0.2 14. ln (4x) � 9.6 15. e8.2 � 10x 16. ln 0.0002 � x�5x � ln 0.2 4x � e9.6 ln 10x � 8.2 ex � 0.0002
Evaluate each expression.
17. ln e3 18. eln 42 19. eln 0.5 20. ln e16.2
3 42 0.5 16.2
1�n
Example 1Example 1
Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 598 Glencoe Algebra 2
Equations and Inequalities with e and ln All properties of logarithms fromearlier lessons can be used to solve equations and inequalities with natural logarithms.
Solve each equation or inequality.
a. 3e2x � 2 � 103e2x � 2 � 10 Original equation
3e2x � 8 Subtract 2 from each side.
e2x � Divide each side by 3.
ln e2x � ln Property of Equality for Logarithms
2x � ln Inverse Property of Exponents and Logarithms
x � ln Multiply each side by �12
�.
x � 0.4904 Use a calculator.
b. ln (4x � 1) � 2
ln (4x � 1) � 2 Original inequality
eln (4x � 1) � e2 Write each side using exponents and base e.
0 � 4x � 1 � e2 Inverse Property of Exponents and Logarithms
1 � 4x � e2 � 1 Addition Property of Inequalities
� x � (e2 � 1) Multiplication Property of Inequalities
0.25 � x � 2.0973 Use a calculator.
Solve each equation or inequality.
1. e4x � 120 2. ex � 25 3. ex � 2 � 4 � 211.1969 {x|x � 3.2189} 4.8332
4. ln 6x � 4 5. ln (x � 3) � 5 � �2 6. e�8x � 50x � 9.0997 17.0855 {x |x � �0.4890}
7. e4x � 1 � 3 � 12 8. ln (5x � 3) � 3.6 9. 2e3x � 5 � 20.9270 6.7196 no solution
10. 6 � 3ex � 1 � 21 11. ln (2x � 5) � 8 12. ln 5x � ln 3x 90.6094 1492.9790 {x |x � 23.2423}
1�4
1�4
8�3
1�2
8�3
8�3
8�3
Study Guide and Intervention (continued)
Base e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
ExampleExample
ExercisesExercises
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 599 Glencoe Algebra 2
Less
on
10-
5
Use a calculator to evaluate each expression to four decimal places.
1. e3 20.0855 2. e�2 0.1353
3. ln 2 0.6931 4. ln 0.09 �2.4079
Write an equivalent exponential or logarithmic equation.
5. ex � 3 x � ln 3 6. e4 � 8x 4 � ln 8x
7. ln 15 � x ex � 15 8. ln x � 0.6931 x � e0.6931
Evaluate each expression.
9. eln 3 3 10. eln 2x 2x
11. ln e�2.5 �2.5 12. ln ey y
Solve each equation or inequality.
13. ex � 5 {x |x � 1.6094} 14. ex � 3.2 {x |x � 1.1632}
15. 2ex � 1 � 11 1.7918 16. 5ex � 3 � 18 1.0986
17. e3x � 30 1.1337 18. e�4x 10 {x |x � �0.5756}
19. e5x � 4 34 {x |x � 0.6802} 20. 1 � 2e2x � �19 1.1513
21. ln 3x � 2 2.4630 22. ln 8x � 3 2.5107
23. ln (x � 2) � 2 9.3891 24. ln (x � 3) � 1 �0.2817
25. ln (x � 3) � 4 51.5982 26. ln x � ln 2x � 2 1.9221
Skills PracticeBase e and Natural Logarithms
© Glencoe/McGraw-Hill 600 Glencoe Algebra 2
Use a calculator to evaluate each expression to four decimal places.
1. e1.5 4.4817 2. ln 8 2.0794 3. ln 3.2 1.1632 4. e�0.6 0.5488
5. e4.2 66.6863 6. ln 1 0 7. e�2.5 0.0821 8. ln 0.037 �3.2968
Write an equivalent exponential or logarithmic equation.
9. ln 50 � x 10. ln 36 � 2x 11. ln 6 � 1.7918 12. ln 9.3 � 2.2300
ex � 50 e2x � 36 e1.7918 � 6 e2.2300 � 9.3
13. ex � 8 14. e5 � 10x 15. e�x � 4 16. e2 � x � 1
x � ln 8 5 � ln 10x x � �ln 4 2 � ln (x � 1)
Evaluate each expression.
17. eln 12 12 18. eln 3x 3x 19. ln e�1 �1 20. ln e�2y �2y
Solve each equation or inequality.
21. ex � 9 22. e�x � 31 23. ex � 1.1 24. ex � 5.8
{x |x � 2.1972} �3.4340 0.0953 1.7579
25. 2ex � 3 � 1 26. 5ex � 1 � 7 27. 4 � ex � 19 28. �3ex � 10 � 8
0.6931 {x |x � 0.1823} 2.7081 {x |x � �0.4055}
29. e3x � 8 30. e�4x � 5 31. e0.5x � 6 32. 2e5x � 24
0.6931 �0.4024 3.5835 0.4970
33. e2x � 1 � 55 34. e3x � 5 � 32 35. 9 � e2x � 10 36. e�3x � 7 � 15
1.9945 1.2036 0 {x |x � �0.6931}
37. ln 4x � 3 38. ln (�2x) � 7 39. ln 2.5x � 10 40. ln (x � 6) � 1
5.0214 �548.3166 8810.5863 8.7183
41. ln (x � 2) � 3 42. ln (x � 3) � 5 43. ln 3x � ln 2x � 9 44. ln 5x � ln x � 7
18.0855 145.4132 36.7493 14.8097
INVESTING For Exercises 45 and 46, use the formula for continuouslycompounded interest, A � Pert, where P is the principal, r is the annual interestrate, and t is the time in years.
45. If Sarita deposits $1000 in an account paying 3.4% annual interest compoundedcontinuously, what is the balance in the account after 5 years? $1185.30
46. How long will it take the balance in Sarita’s account to reach $2000? about 20.4 yr
47. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after t years is given by the equation y � aekt, where a is the initial amount present and k isthe decay constant for the radioactive substance. If a � 100, y � 50, and k � �0.035,find t. about 19.8 yr
Practice (Average)
Base e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
Reading to Learn MathematicsBase e and Natural Logarithms
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 601 Glencoe Algebra 2
Less
on
10-
5
Pre-Activity How is the natural base e used in banking?
Read the introduction to Lesson 10-5 at the top of page 554 in your textbook.
Suppose that you deposit $675 in a savings account that pays an annualinterest rate of 5%. In each case listed below, indicate which method ofcompounding would result in more money in your account at the end of oneyear.a. annual compounding or monthly compounding monthlyb. quarterly compounding or daily compounding dailyc. daily compounding or continuous compounding continuous
Reading the Lesson1. Jagdish entered the following keystrokes in his calculator:
5
The calculator returned the result 1.609437912. Which of the following conclusions arecorrect? d and fa. The common logarithm of 5 is about 1.6094.
b. The natural logarithm of 5 is exactly 1.609437912.
c. The base 5 logarithm of e is about 1.6094.
d. The natural logarithm of 5 is about 1.609438.
e. 101.609437912 is very close to 5.
f. e1.609437912 is very close to 5.
2. Match each expression from the first column with its value in the second column. Somechoices may be used more than once or not at all.
a. eln 5 IV I. 1
b. ln 1 V II. 10
c. eln e VI III. �1
d. ln e5 IV IV. 5
e. ln e I V. 0
f. ln � � III VI. e
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose that you arestudying with a classmate who is puzzled when asked to evaluate ln e3. How would youexplain to him an easy way to figure this out? Sample answer: ln means naturallog. The natural log of e3 is the power to which you raise e to get e3. Thisis obviously 3.
1�e
ENTER) LN
© Glencoe/McGraw-Hill 602 Glencoe Algebra 2
Approximations for � and eThe following expression can be used to approximate e. If greater and greatervalues of n are used, the value of the expression approximates e more andmore closely.
�1 � �n1
��n
Another way to approximate e is to use this infinite sum. The greater thevalue of n, the closer the approximation.
e � 1 � 1 � �12� � �2
1� 3� � �2 �
13 � 4� � … � �2 � 3 � 4
1� … � n� � …
In a similar manner, � can be approximated using an infinite productdiscovered by the English mathematician John Wallis (1616–1703).
��2� � �
21� � �
23� � �
43� � �
45� � �
65� � �
67� � … � �2n
2�n
1� � �2n2�n
1� …
Solve each problem.
1. Use a calculator with an ex key to find e to 7 decimal places.
2. Use the expression �1 � �n1
��nto approximate e to 3 decimal places. Use
5, 100, 500, and 7000 as values of n.
3. Use the infinite sum to approximate e to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.
4. Which approximation method approaches the value of e more quickly?
5. Use a calculator with a � key to find � to 7 decimal places.
6. Use the infinite product to approximate � to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.
7. Does the infinite product give good approximations for � quickly?
8. Show that � 4 � � 5 is equal to e6 to 4 decimal places.
9. Which is larger, e� or � e?
10. The expression x reaches a maximum value at x � e. Use this fact to prove the inequality you found in Exercise 9.
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
Study Guide and InterventionExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 603 Glencoe Algebra 2
Less
on
10-
6Exponential Decay Depreciation of value and radioactive decay are examples ofexponential decay. When a quantity decreases by a fixed percent each time period, theamount of the quantity after t time periods is given by y � a(1 � r)t, where a is the initialamount and r is the percent decrease expressed as a decimal.Another exponential decay model often used by scientists is y � ae�kt, where k is a constant.
CONSUMER PRICES As technology advances, the price of manytechnological devices such as scientific calculators and camcorders goes down.One brand of hand-held organizer sells for $89.
a. If its price decreases by 6% per year, how much will it cost after 5 years?Use the exponential decay model with initial amount $89, percent decrease 0.06, andtime 5 years.y � a(1 � r)t Exponential decay formula
y � 89(1 � 0.06)5 a � 89, r � 0.06, t � 5
y � $65.32After 5 years the price will be $65.32.
b. After how many years will its price be $50?To find when the price will be $50, again use the exponential decay formula and solve for t.
y � a(1 � r)t Exponential decay formula
50 � 89(1 � 0.06)t y � 50, a � 89, r � 0.06
� (0.94)t Divide each side by 89.
log � � � log (0.94)t Property of Equality for Logarithms
log � � � t log 0.94 Power Property
t � Divide each side by log 0.94.
t � 9.3The price will be $50 after about 9.3 years.
1. BUSINESS A furniture store is closing out its business. Each week the owner lowersprices by 25%. After how many weeks will the sale price of a $500 item drop below $100?6 weeks
CARBON DATING Use the formula y � ae�0.00012t, where a is the initial amount ofCarbon-14, t is the number of years ago the animal lived, and y is the remainingamount after t years.
2. How old is a fossil remain that has lost 95% of its Carbon-14? about 25,000 years old
3. How old is a skeleton that has 95% of its Carbon-14 remaining? about 427.5 years old
log ��5809��
��log 0.94
50�89
50�89
50�89
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 604 Glencoe Algebra 2
Exponential Growth Population increase and growth of bacteria colonies are examplesof exponential growth. When a quantity increases by a fixed percent each time period, theamount of that quantity after t time periods is given by y � a(1 � r)t, where a is the initialamount and r is the percent increase (or rate of growth) expressed as a decimal.Another exponential growth model often used by scientists is y � aekt, where k is a constant.
A computer engineer is hired for a salary of $28,000. If she gets a5% raise each year, after how many years will she be making $50,000 or more?Use the exponential growth model with a � 28,000, y � 50,000, and r � 0.05 and solve for t.
y � a(1 � r)t Exponential growth formula
50,000 � 28,000(1 � 0.05)t y � 50,000, a � 28,000, r � 0.05
� (1.05)t Divide each side by 28,000.
log � � � log (1.05)t Property of Equality of Logarithms
log � � � t log 1.05 Power Property
t � Divide each side by log 1.05.
t � 11.9 years Use a calculator.
If raises are given annually, she will be making over $50,000 in 12 years.
1. BACTERIA GROWTH A certain strain of bacteria grows from 40 to 326 in 120 minutes.Find k for the growth formula y � aekt, where t is in minutes. about 0.0175
2. INVESTMENT Carl plans to invest $500 at 8.25% interest, compounded continuously.How long will it take for his money to triple? about 14 years
3. SCHOOL POPULATION There are currently 850 students at the high school, whichrepresents full capacity. The town plans an addition to house 400 more students. If the school population grows at 7.8% per year, in how many years will the new additionbe full? about 5 years
4. EXERCISE Hugo begins a walking program by walking mile per day for one week.
Each week thereafter he increases his mileage by 10%. After how many weeks is hewalking more than 5 miles per day? 24 weeks
5. VOCABULARY GROWTH When Emily was 18 months old, she had a 10-wordvocabulary. By the time she was 5 years old (60 months), her vocabulary was 2500 words.If her vocabulary increased at a constant percent per month, what was that increase?about 14%
1�2
log ��5208��
�log 1.05
50�28
50�28
50�28
Study Guide and Intervention (continued)
Exponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
ExampleExample
ExercisesExercises
Skills PracticeExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 605 Glencoe Algebra 2
Less
on
10-
6Solve each problem.
1. FISHING In an over-fished area, the catch of a certain fish is decreasing at an averagerate of 8% per year. If this decline persists, how long will it take for the catch to reachhalf of the amount before the decline? about 8.3 yr
2. INVESTING Alex invests $2000 in an account that has a 6% annual rate of growth. Tothe nearest year, when will the investment be worth $3600? 10 yr
3. POPULATION A current census shows that the population of a city is 3.5 million. Usingthe formula P � aert, find the expected population of the city in 30 years if the growthrate r of the population is 1.5% per year, a represents the current population in millions,and t represents the time in years. about 5.5 million
4. POPULATION The population P in thousands of a city can be modeled by the equationP � 80e0.015t, where t is the time in years. In how many years will the population of thecity be 120,000? about 27 yr
5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to50,000 if the growth rate per day is 93.2%? about 4.9 days
6. NUCLEAR POWER The element plutonium-239 is highly radioactive. Nuclear reactorscan produce and also use this element. The heat that plutonium-239 emits has helped topower equipment on the moon. If the half-life of plutonium-239 is 24,360 years, what isthe value of k for this element? about 0.00002845
7. DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite informationto locate ground position. Abu’s surveying firm bought a GPS system for $12,500. TheGPS depreciated by a fixed rate of 6% and is now worth $8600. How long ago did Abubuy the GPS system? about 6.0 yr
8. BIOLOGY In a laboratory, an organism grows from 100 to 250 in 8 hours. What is thehourly growth rate in the growth formula y � a(1 � r) t? about 12.13%
© Glencoe/McGraw-Hill 606 Glencoe Algebra 2
Solve each problem.
1. INVESTING The formula A � P�1 � �2tgives the value of an investment after t years in
an account that earns an annual interest rate r compounded twice a year. Suppose $500is invested at 6% annual interest compounded twice a year. In how many years will theinvestment be worth $1000? about 11.7 yr
2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to2000 if the growth rate per hour is 85%? about 7.5 h
3. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find theconstant k in the decay formula for the substance. about 0.02166
4. DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fixed rate of12% per year. After how many years will the value have depreciated to $100,000?about 7.2 yr
5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 yearsago would cost $12,500. Since Dave bought the car, the inflation rate for cars like his hasbeen at an average annual rate of 5.1%. If Dave originally paid $8400 for the car, howlong ago did he buy it? about 8 yr
6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used totrace the path of nonradioactive substances in a system. What is the value of k forCobalt-60? about 0.1216
7. WHALES Modern whales appeared 5�10 million years ago. The vertebrae of a whalediscovered by paleontologists contain roughly 0.25% as much carbon-14 as they wouldhave contained when the whale was alive. How long ago did the whale die? Use k � 0.00012. about 50,000 yr
8. POPULATION The population of rabbits in an area is modeled by the growth equationP(t) � 8e0.26t, where P is in thousands and t is in years. How long will it take for thepopulation to reach 25,000? about 4.4 yr
9. DEPRECIATION A computer system depreciates at an average rate of 4% per month. Ifthe value of the computer system was originally $12,000, in how many months is itworth $7350? about 12 mo
10. BIOLOGY In a laboratory, a culture increases from 30 to 195 organisms in 5 hours.What is the hourly growth rate in the growth formula y � a(1 � r) t? about 45.4%
r�2
Practice (Average)
Exponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
Reading to Learn MathematicsExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 607 Glencoe Algebra 2
Less
on
10-
6Pre-Activity How can you determine the current value of your car?
Read the introduction to Lesson 10-6 at the top of page 560 in your textbook.
• Between which two years shown in the table did the car depreciate bythe greatest amount?between years 0 and 1
• Describe two ways to calculate the value of the car 6 years after it waspurchased. (Do not actually calculate the value.)Sample answer: 1. Multiply $9200.66 by 0.16 and subtract theresult from $9200.66. 2. Multiply $9200.66 by 0.84.
Reading the Lesson
1. State whether each situation is an example of exponential growth or decay.
a. A city had 42,000 residents in 1980 and 128,000 residents in 2000. growth
b. Raul compared the value of his car when he bought it new to the value when hetraded ‘;lpit in six years later. decay
c. A paleontologist compared the amount of carbon-14 in the skeleton of an animalwhen it died to the amount 300 years later. decay
d. Maria deposited $750 in a savings account paying 4.5% annual interest compoundedquarterly. She did not make any withdrawals or further deposits. She compared thebalance in her passbook immediately after she opened the account to the balance 3 years later. growth
2. State whether each equation represents exponential growth or decay.
a. y � 5e0.15t growth b. y � 1000(1 � 0.05) t decay
c. y � 0.3e�1200t decay d. y � 2(1 � 0.0001) t growth
Helping You Remember
3. Visualizing their graphs is often a good way to remember the difference betweenmathematical equations. How can your knowledge of the graphs of exponential equationsfrom Lesson 10-1 help you to remember that equations of the form y � a(1 � r) t
represent exponential growth, while equations of the form y � a(1 � r) t representexponential decay?Sample answer: If a � 0, the graph of y � abx is always increasing if b � 1 and is always decreasing if 0 � b � 1. Since r is always a positivenumber, if b � 1 � r, the base will be greater than 1 and the function willbe increasing (growth), while if b � 1 � r, the base will be less than 1and the function will be decreasing (decay).
© Glencoe/McGraw-Hill 608 Glencoe Algebra 2
Effective Annual YieldWhen interest is compounded more than once per year, the effective annualyield is higher than the annual interest rate. The effective annual yield, E, isthe interest rate that would give the same amount of interest if the interestwere compounded once per year. If P dollars are invested for one year, thevalue of the investment at the end of the year is A � P(1 � E). If P dollarsare invested for one year at a nominal rate r compounded n times per year,
the value of the investment at the end of the year is A � P�1 � �nr
��n. Setting
the amounts equal and solving for E will produce a formula for the effectiveannual yield.
P(1 � E) � P�1 � �nr
��n
1 � E � �1 � �nr
��n
E � �1 � �nr
��n� 1
If compounding is continuous, the value of the investment at the end of oneyear is A � Per. Again set the amounts equal and solve for E. A formula forthe effective annual yield under continuous compounding is obtained.
P(1 � E) � Per
1 � E � er
E � er � 1
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
Find the effectiveannual yield of an investment made at7.5% compounded monthly.r � 0.075
n � 12
E � �1 � �0.
10275��12
� 1 � 7.76%
Find the effectiveannual yield of an investment made at6.25% compounded continuously.r � 0.0625
E � e0.0625 � 1 � 6.45%
Example 1Example 1 Example 2Example 2
Find the effective annual yield for each investment.
1. 10% compounded quarterly 2. 8.5% compounded monthly
3. 9.25% compounded continuously 4. 7.75% compounded continuously
5. 6.5% compounded daily (assume a 365-day year)
6. Which investment yields more interest—9% compounded continuously or 9.2% compounded quarterly?
Chapter 10 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 609 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find the domain and range of the function whose graph is shown.A. D � {x � x � 0}; R � {y � y � 0}B. D � {x � x is any real number.}; R � { y � y � 0}C. D � {x � x � 0}; R � {y � y is any real number.}D. D � {x � x is any real number.}; R � {y � y � 0} 1.
2. Which function represents exponential growth?
A. y � 9��13��
xB. y � 4x4 C. y � 12��
15��
xD. y � 10(2)x 2.
3. The graph of which exponential function passes through the points (0, 4) and (1, 24)?A. y � 4(6)x B. y � 3(8)x C. y � 2(2)x D. y � 10(3)x 3.
4. Simplify (x�7�)�3�.
A. x�21� B. x�7� � �3� C. x�10� D. x��73�� 4.
5. Solve 23m � 4 � 4.
A. m � 0 B. m � 0 C. m � 2 D. m � �53� 5.
6. Write the equation 43 � 64 in logarithmic form.A. log4 3 � 64 B. log3 4 � 64 C. log64 4 � 3 D. log4 64 � 3 6.
7. Write the equation log12 144 � 2 in exponential form.A. 1442 � 12 B. 122 � 144 C. 212 � 144 D. 14412 � 2 7.
8. Evaluate log2 8.A. 3 B. 4 C. 16 D. 64 8.
9. Solve log3 n � 2.A. 6 B. 5 C. 8 D. 9 9.
10. Solve log2 2m � log2 (m � 5).
A. m � �53� B. m � 5 C. m � 5 D. m � �5 10.
1010
y
xO
y � 4(2)x
© Glencoe/McGraw-Hill 610 Glencoe Algebra 2
Chapter 10 Test, Form 1 (continued)
11. Use log5 2 � 0.4307 to approximate the value of log5 4.A. 0.8614 B. 0.8980 C. 1.3652 D. 0.1855 11.
12. Solve log6 10 � log6 x � log6 40.A. 180 B. 4 C. 5 D. 30 12.
13. Solve 4x � 20. Round to four decimal places.A. 0.4628 B. 1.5214 C. 0.6990 D. 2.1610 13.
14. Solve 3x � 21. Round to four decimal places.A. x � 0.8451 B. x � 2.7712 C. x � 0.3608 D. x � 7.0000 14.
15. Express log9 22 in terms of common logarithms.
A. log �292� B. log 198 C. �
lloogg
292
� D. �lloogg
292� 15.
16. Evaluate eln 4.A. e4 B. 4e C. ln 4 D. 4 16.
17. Solve ex � 2.7.A. x � 0.9933 B. x � 0.9933 C. x � 1.0668 D. x � 1.0668 17.
18. Solve ln 3x � 1.A. 20.0855 B. 0.3333 C. 0.9061 D. 8.1548 18.
19. AUTOMOBILES Lydia bought a car for $20,000. It is expected to depreciate at a rate of 10% per year. What will be the value of the car in 2 years? Use y � a(1 � r)t and round to the nearest dollar.A. $16,200 B. $16,000 C. $19,980 D. $18,050 19.
20. ART Martin bought a painting for $5,000. It is expected to appreciate at 4% per year. How much will the painting be worth in 6 years? Use y � a(1 � r)t and round to the nearest cent.A. $6200.00 B. $5360.38 C. $37,647.68 D. $6326.60 20.
Bonus Evaluate 3 log2 64 � eln 5 � log�13�
9. B:
NAME DATE PERIOD
1010
Chapter 10 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 611 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find the domain and range of the function y � 3��15��
x.
A. D � {x � x is any real number.} B. D � {x � x is any real number.}R � { y � y � 0} R � { y � y � 0}
C. D � {x � x � 0} D. D � {x � x � 0}R � { y � y � 0} R � { y � y is any real number.} 1.
2. Which function represents exponential decay?
A. y � �1100�
(6)x B. y � (4x)�12�
C. y � 2��43��
xD. y � 12��
18��
x2.
3. Use the equation of the exponential function whose graph passes through the points (0, �3) and (2, �48) to find the value of y when x � �2.
A. ��34� B. ��
38� C. ��1
36�
D. 48 3.
4. Simplify m9�5� � m�5�.
A. m45 B. m9 C. m8�5� D. m10�5� 4.
5. Solve ��316��
n� 216n � 5.
A. 10 B. 3 C. �3 D. �10 5.
6. Solve 81y � 27y � 3
A. y � �9 B. y � 9 C. y � �9 D. y � 9 6.
7. Write the equation 6561�14�
� 9 in logarithmic form.
A. log�14�
9 � 6561 B. log6561 9 � �14�
C. log9 6561 � �14� D. log
�14�
6561 � 9 7.
8. Evaluate 5log5 63.A. 58 B. 315 C. log5 63 D. 63 8.
9. Solve log�15
�x � �1.
A. �215�
B. �5 C. 5 D. ��15� 9.
10. Solve log3 (5x � 1) � log3 (3x � 7)A. x � 3 B. x � 4 C. x 6 D. x � 27 10.
1010
© Glencoe/McGraw-Hill 612 Glencoe Algebra 2
Chapter 10 Test, Form 2A (continued)
11. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of log5 54.A. 0.1370 B. 2.4785 C. 0.8820 D. 0.7488 11.
12. Solve log4 (m � 3) � log4 (m � 3) � 2.A. �11� B. 5 C. 1 D. �5.5 12.
13. Solve 63n � 435n � 4. Round to four decimal places.A. 1.1202 B. �1.9005 C. �0.2800 D. 2.1418 13.
14. Solve 52x � 1 � 50. Round to four decimal places.A. x � 4.5000 B. x � 0.7153 C. x � 0 D. x � 2.4307 14.
15. Use common logarithms to approximate log9 207 to four decimal places.A. 0.4120 B. 1.3617 C. 3.2702 D. 2.4270 15.
16. Evaluate ln e�9x.A. �9 ln x B. 9 ln x C. �9x D. 9x 16.
17. Solve 4 � 3e5x � 27.A. 0.4074 B. 0.4394 C. 2.0369 D. 0.1769 17.
18. Solve ln (x � 5) � 2.A. x � 2.3891 B. x 2.3891 C. x � 12.3891 D. x 12.3891 18.
19. CHEMISTRY A particular compound decays according to the equation y � ae�0.0974t, where t is in days. Find the half-life of this compound.A. about 5.1 days B. about 7.4 daysC. about 7.1 days D. about 9.7 days 19.
20. TOURISM At a town with a large convention center, the cost of a hotel room has increased 5.1% annually. If the average hotel room cost $48.00 in 1980 and this growth continues, what will an average hotel room cost in 2012? Use y � a(1 � r)t and round to the nearest cent.A. $143.38 B. $235.79 C. $126.34 D. $87.19 20.
Bonus Solve 5log5 2x � log5 (x � 3) � ln ex � 4. B:
NAME DATE PERIOD
1010
Chapter 10 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 613 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find the domain and range of the function y � �12�(2)x.
A. D � {x � x is any real number.} B. D � {x � x is any real number.}R � { y � y � 0} R � { y � y � 0}
C. D � {x � x � 0} D. D � {x � x � 0}R � { y � y � 0} R � { y � y is any real number.} 1.
2. Which function represents exponential growth?
A. y � �210���
52��
xB. y � 16(0.4)x C. y � �
12���
18��
xD. y � 8x3 2.
3. Use the equation of the exponential function whose graph passes through the points (0, �2) and (2, �50) to find the value of y when x � �2.
A. ��1100�
B. 50 C. ��225�
D. ��510�
3.
4. Simplify s7�11� � s�11�.
A. s77 B. s6�11� C. s8�11� D. s7 4.
5. Solve ��811��
t� 243t � 2.
A. �92� B. �
190� C. �
29� D. �1
90�
5.
6. Solve 64x � 32x � 2.A. x � �10 B. x � �10 C. x � 10 D. x � 10 6.
7. Write the equation log243 81 � �45� in exponential form.
A. 81�45�
� 243 B. 243�45�
� 81 C. ��45��
81� 243 D. ��
45��
243� 81 7.
8. Evaluate 9log9 54.A. log9 54 B. 54 C. 6 D. 486 8.
9. Solve log�18�
x � �1.
A. 8 B. �8 C. 0 D. ��18� 9.
10. Solve log2 (7x � 3) � log2 (x � 12).
A. x �52� B. x ��
52� C. x � �
32� D. x � �
52� 10.
1010
© Glencoe/McGraw-Hill 614 Glencoe Algebra 2
Chapter 10 Test, Form 2B (continued)
11. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of log5 12.A. 0.8681 B. 0.1266 C. 1.5440 D. 0.5880 11.
12. Solve log3 a � log3 (a � 8) � 2.A. 8 B. 5 C. 9 D. �1, 9 12.
13. Solve 92n � 404n � 7. Round to four decimal places.A. 2.4922 B. 0.4012 C. �0.3560 D. �4.7209 13.
14. Solve 35x � 1 30. Round to four decimal places.A. x 0.4000 B. x 0.8192 C. x 1.8000 D. x 3.0959 14.
15. Use common logarithms to approximate log7 448 to four decimal places.A. 0.3188 B. 1.8062 C. 3.1372 D. �1.8062 15.
16. Evaluate eln (�6x).A. �6x B. 6 ln x C. 6x D. �6 ln x 16.
17. Solve ln (x � 2) � 3.A. 22.0855 B. 18.0855 C. 20.0855 D. �0.9014 17.
18. Solve e�9x 6.A. x � �1.8122 B. x 1.7918C. x � �0.08646 D. x � �0.1991 18.
19. CHEMISTRY A particular compound decays according to the equation y � ae�0.0736t, where t is in days. Find the half-life of the compound.A. about 9.1 days B. about 9.4 daysC. about 6.8 days D. about 7.4 days 19.
20. FOOD PRICES At a wholesale food distribution center, the price of sugar has increased 6.3% annually since 1980. Suppose sugar cost $0.43 per pound in 1980 and this growth continues. What will a pound of sugar cost in 2017? Use y � a(1 � r)t and round to the nearest cent.A. $4.12 B. $1.21 C. $2.42 D. $3.30 20.
Bonus Solve 2log2 5x � log2 (x � 1) � ln ex. B:
NAME DATE PERIOD
1010
Chapter 10 Test, Form 2C
© Glencoe/McGraw-Hill 615 Glencoe Algebra 2
1. Sketch the graph of y � �12�(3)x. Then state the function’s 1.
domain and range.
2. Determine whether the function y � 0.8��23��
xrepresents 2.
exponential growth or decay.
3. Write an exponential function whose graph passes through 3.the points (0, �6) and (�2, �54).
4. Simplify z5�7� z�7�. 4.
5. Solve �16� � 6n � 4. 5.
6. Solve 32x � 16x � 2. 6.
7. Write the equation log81 �19� � ��
12� in exponential form. 7.
8. Evaluate log9 97. 8.
9. Evaluate log4 128. 9.
10. Solve log36 n � �32�. 10.
11. Solve log5 (8x) � log5 (3x � 10). 11.
Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of each expression.
12. log5 48 12.
13. log5 �53� 13.
y
xO
NAME DATE PERIOD
SCORE 1010
Ass
essm
ent
© Glencoe/McGraw-Hill 616 Glencoe Algebra 2
Chapter 10 Test, Form 2C (continued)
For Questions 14–19, solve each equation or inequality.If necessary, round to four decimal places.
14. log4 n � �14� log4 81 � �
12� log4 25 14.
15. log2 (2x � 6) � log2 x � 3 15.
16. log3 (x � 3) � log3 (x � 2) � log3 14 16.
17. 6n � 2 � 50 17.
18. 2y � 5 y � 2 18.
19. 43x � 1 � 28 19.
20. Express log12 4 in terms of common logarithms. Then 20.approximate its value to four decimal places.
21. Evaluate ln e30. 21.
22. Solve ln (x � 5) � 3. 22.
23. Solve e�4x 9. 23.
24. CHEMISTRY After 12 hours, half of a 16-gram sample of a 24.radio-active element remains. Find the constant k for this element for t hours, then write the equation for modeling its exponential decay.
25. SAVINGS A savings account deposit of $150 is to earn 6.5% 25.interest. After how many years will the investment be worth $450? Use y � a(1 � r)t and round to the nearest tenth.
Bonus Evaluate (log4 123)(log12 43). B:
NAME DATE PERIOD
1010
Chapter 10 Test, Form 2D
© Glencoe/McGraw-Hill 617 Glencoe Algebra 2
1. Sketch the graph of y � 6��12��
x. Then state the function’s 1.
domain and range.
2. Determine whether the function y � 0.3��85��
xrepresents 2.
exponential growth or decay.
3. Write an exponential function whose graph passes through 3.the points (0, �5) and (�3, �40).
4. Simplify m9�6� � m�6�. 4.
5. Solve ��14��
m � 7� 16. 5.
6. Solve 10x � 3 � 100x � 1. 6.
7. Write the equation 5�4 � �6125�
in logarithmic form. 7.
8. Evaluate log6 68. 8.
9. Evaluate log8 128. 9.
10. Solve log64 x � �23�. 10.
11. Solve log4 (2x � 5) log4 (3x � 2). 11.
Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of each expression.
12. log5 18 12.
13. log5 �52� 13.
y
xO
NAME DATE PERIOD
SCORE 1010
Ass
essm
ent
© Glencoe/McGraw-Hill 618 Glencoe Algebra 2
Chapter 10 Test, Form 2D (continued)
For Questions 14–19, solve each equation or inequality.If necessary, round to four decimal places.
14. log5 n � �13� log5 64 � �
12� log5 49 14.
15. log6 (5 � 2a) � log6 (3a) � 1 15.
16. log3 (x � 3) � log3 (x � 2) � log3 6 16.
17. 7n � 3 � 80 17.
18. 3n � 6n � 2 18.
19. 54x � 1 � 30 19.
20. Express log15 5 in terms of common logarithms. Then 20.approximate its value to four decimal places.
21. Evaluate eln 22. 21.
22. Solve ln (x � 4) � 4. 22.
23. Solve e�3x 18. 23.
24. CHEMISTRY In 5 years, radioactivity reduces the mass of 24.a 100-gram sample of an element to 75 grams. Find the constant k for this element for t in years, then write the equation for modeling this exponential decay.
25. SAVINGS A savings account deposit of $300 is to earn 5.8% 25.interest. After how many years will the investment be worth $900? Use y � a(1 � r)t and round to the nearest tenth.
Bonus Evaluate (log5 204)(log20 54). B:
NAME DATE PERIOD
1010
Chapter 10 Test, Form 3
© Glencoe/McGraw-Hill 619 Glencoe Algebra 2
1. Sketch the graph of y � �1.5(4)x. Then state the function’s 1.domain and range.
2. Determine whether the function y � 0.4(3.8)�x represents 2.exponential growth or decay.
3. Write an exponential function whose graph passes through 3.(0, �0.3) and (2, �10.8)
4. Solve 24x 321 � x � 8x � 2. 4.
5. Solve ��811��
4m � 1� ��2
17��
5m. 5.
6. Evaluate 2log2 (8x � 1). 6.
7. Evaluate log3 243x. 7.
8. Solve logx [log2 (log3 9)] � 2. 8.
9. Solve log3 (a2 � 12) � log3 a. 9.
For Questions 10 and 11, use log5 2 � 0.4307 andlog5 3 � 0.6826 to approximate the value of each expression.
10. log5 �145� 10.
11. log5 1.2 11.
12. Solve log4 0.25 � 3 log4 x � 5 log4 2 � �13� log4 64. 12.
13. Solve log4 (4b � 14) � log4 (b2 � 3b � 17) � �12�. 13.
y
xO
NAME DATE PERIOD
SCORE 1010
Ass
essm
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© Glencoe/McGraw-Hill 620 Glencoe Algebra 2
Chapter 10 Test, Form 3 (continued)
14. Solve loga (3n) � 2 loga x � loga x for n. 14.
15. Solve 4.5x2 � 2 � 32.7. Round to four decimal places. 15.
16. Solve 3n � �5n � 2�. Round to four decimal places. 16.
17. Solve ��12��
2t� 53 � 4t. Round to four decimal places. 17.
18. Express log5 (2.1)3 in terms of common logarithms. Then 18.approximate its value to four decimal places.
19. Evaluate e4 ln x. 19.
20. Solve ln (x � 3) � ln x � ln 4. 20.
21. Solve ln (x2 � 10) � ln x � ln 7. 21.
POPULATIONS For Questions 22 and 23, use the following information.
The population of Suffolk County in Massachusetts decreased from 663,906 in 1990 to 641,695 in 1999.
22. Write an exponential decay equation of the form y � aekt for 22.Suffolk County, where t is the number of years after 1990.
23. Use your equation to predict the population of Suffolk 23.County in 2020.
24. HOME OWNERSHIP The Richardson family bought a house 24.12 years ago for $95,000. The house is now worth $167,000.Assuming a steady rate of growth, what was the yearly rate of appreciation?
25. SCHOOL ENROLLMENT At a certain school, the number of 25.children entering kindergarten increased by 6.7% annually for 5 years and then decreased by 4.2% annually in the next 5 years. If 110 children enrolled in kindergarten at the beginning of this period, how many were enrolled after 10 years?
Bonus Solve log x2 � (log x)2. B:
NAME DATE PERIOD
1010
Chapter 10 Open-Ended Assessment
© Glencoe/McGraw-Hill 621 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.
1. For the equation y � abx, where a � 0, we know that if b � 1, the function represents exponential growth, while it represents exponential decay if 0 � b � 1.a. Choose a positive value for a and let b � 1. Complete the table for these values
of a and b. Is y � abx an exponential function? Explain your reasoning.
b. Choose a positive value for a and a negative value for b. Complete the table for these values. Is y � abx an exponential function? Explain your reasoning.
2. a. Solve the exponential equation 35x � 9x � 6 by rewriting the equation so that each side has the same base.
b. Solve the equation in part a using common logarithms.c. Which method do you prefer? Explain your reasoning.d. Write and solve an exponential equation that you would also prefer to
solve using the method you chose in part c.
3. a. How are the three equations below alike? How are they different?log3 x � 2 log x � 2 ln x � 2
b. Solve each equation in part a above. Then write and solve a fourth equation that shares the same similarities and differences as the three given equations.
4. Ruby solved the exponential inequality 22z � 12z � 1 and stated that the solution set was {z � z � �2.2619}. When she checked her solution however, Ruby found that z � 1, which is in her solution set, does not make the original inequality true. When shechecked z � �3, which is not in her solution set, the original inequality is true.a. Show how Ruby arrived at her solution using common logarithms.b. What do her checks of z � 1 and z � �3 indicate about Ruby’s solution?c. What change must be made to the solution and why must that change be made?
5. ECONOMICS The Jones Corporation found that its annual profit could bemodeled by the exponential equation y � 10(0.99)t, while the Davis Company’sannual profit is modeled by y � 8(1.01)t. For both equations, profit is given inmillions of dollars, and t is the number of years since 1990.a. Find each company’s annual profit for the years between 1990 and 2000
to the nearest dollar.b. In which company would you prefer to own stock? Explain your reasoning.c. Indicate how a comparison of the two profit equations would support
your decision.
NAME DATE PERIOD
SCORE 1010
Ass
essm
ent
x �3 �2 �1 0 1 2 3
y � abx
x �3 �2 �1 0 1 2 3
y � abx
© Glencoe/McGraw-Hill 622 Glencoe Algebra 2
Chapter 10 Vocabulary Test/Review
Choose from the terms above to complete each sentence.
1. A logarithm with base e is called a(n) .
2. The function y � 10x is an example of a(n) .
3. The equation y � e�0.2t is a model for .
4. The inverse of the function y � ex is the
5. The equation y � 100(1 � 0.1)t is a model for .
6. The value of log3 50 can be found by using the with a calculator.
7. y � log2 x is an example of a .
8. 5x � 1 � 125 and 9x � 272x � 1 are examples of .
9. A logarithm with base 10 is called a(n) .
10. In the equation y � 20(1 � 0.02)t, 0.02 is called the .
In your own words—Define each term.
11. logarithm
12. natural base, e
Change of Base Formulacommon logarithmexponential decayexponential equationexponential function
exponential growthexponential inequalitylogarithmlogarithmic function
natural base, enatural base exponential function
natural logarithm
natural logarithmic function
rate of decayrate of growth
NAME DATE PERIOD
SCORE 1010
Ass
essm
ent
Chapter 10 Quiz (Lessons 10–1 and 10–2)
1010
© Glencoe/McGraw-Hill 623 Glencoe Algebra 2
1. Sketch the graph of y � 3��12��
x. Then state the function’s
domain and range.
2. Write an exponential function whose graph passes through the points (0, �5) and (�2, �20). Then determine whether the function represents exponential growth or decay.
3. Simplify 3�5� 32�5�. 3.
4. Solve ��13��
m� 27m � 2. 4.
5. Solve 254t � 1 � 1252t. 5.
6. Write the equation 81�12�
� 9 in logarithmic form. 6.
7. Write the equation log216 36 � �23� in exponential form. 7.
8. Evaluate log16 64. 8.
9. Solve log16 n � ��12�. 9.
10. Solve log5 (4x � 1) � log5 (x � 2). 10.
NAME DATE PERIOD
SCORE
Chapter 10 Quiz (Lesson 10–3)
Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the value of each expression. 1.
1. log5 �130� 2. log5 24 2.
Solve each equation.
3. log7 36 � log7 (2x) � log7 4 3.
4. log3 x � �12� log3 25 � 5 log3 2 4.
5. log2 (x � 1) � log2 (x � 5) � 4 5.
NAME DATE PERIOD
SCORE 1010
1. y
xO
© Glencoe/McGraw-Hill 624 Glencoe Algebra 2
Use a calculator to evaluate each expression to four decimal places.
1. log 1.5 2. ln 4.1
For Questions 3–7, solve each equation or inequality.Round to four decimal places.
3. 42m � 130 4. 5x � 4 � 23x
5. 7t � 5 � 21.5 6. ln (x � 5) � 3
7. 4 � 2e5x � 28 7.
8. Express log3 25 in terms of common logarithms. Then approximate its value to four decimal places. 8.
9. Write an equivalent logarithmic equation for e3 � 2x. 9.
10. Evaluate eln 0.3. 10.
Chapter 10 Quiz (Lesson 10–6)
1. A substance decays according to the equation y � ae�0.0025t, 1.where t is in minutes. Find the half-life of the substance.Round to the nearest tenth.
2. A-1 Electric has a piece of machinery valued at $55,000. 2.It depreciates at a rate of 12.5% per year. After how many years will the value have depreciated to $38,000? Round to the nearest tenth.
3. Standardized Test Practice In 1925, the population of acity was 90,000. Since then, the population has increased by 2.1% per year. If it continues to grow at this rate, what will the population be in 2020? A. 4,073,333 B. 136,382 C. 648,169 D. 6.6 � 1012 3.
4. The Morgans bought a house worth $125,000. Assuming 4.that the house will appreciate 8% per year, what will the house be worth in eight years? Round to the nearest dollar.
5. A type of bacteria doubles in number every 25 minutes. 5.Find the constant k for this type of bacteria, then write the equation for modeling this exponential growth.
NAME DATE PERIOD
SCORE
Chapter 10 Quiz (Lessons 10–4 and 10–5)
1010
NAME DATE PERIOD
SCORE
1010
1.
2.
3.
4.
5.
6.
Chapter 10 Mid-Chapter Test (Lessons 10–1 through 10–4)
© Glencoe/McGraw-Hill 625 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. Find the domain and range of the function shown.A. D � {x � x � 0}, R � { y � y is any real number.}B. D � {x � x is any real number.}, R � { y � y � 0}C. D � {x � x is any real number.}, R � { y � y � 0}D. D � {x � x � 0}, R � { y � y � 0} 1.
2. Simplify the expression y5�7� � y�7�.
A. y35 B. y5 C. y6�7� D. y4�7� 2.
3. Write the equation 4�3 � �614�
in logarithmic form.
A. log�3 4 � �614�
B. log4 �614�
� �3
C. log�614�
(�3) � 4 D. log4 (�3) � �614�
3.
4. Evaluate log4 32.
A. �52� B. 8 C. 3 D. �
25� 4.
5. Solve log3 (7x � 3) � log3 (5x).
A. x � �32� B. x � �
37� C. x � 0 D. x � �
23� 5.
6. Use log3 5 � 1.4650 and log3 7 � 1.7712 to approximate the value of log3 �251�.
A. 3.6270 B. 3.8486 C. 1.8916 D. 1.3062 6.
7. Write an exponential function whose graph passes through 7.the points (0, �3) and (4, �48).
For Questions 9–13, solve each equation or inequality. 8.
8. ��18��
x� 4x � 5 9. log
�15�
m � �2 9.
10. log7 (x � 3) � log7 (x � 3) � 1 10.
11. log3 (y � 8) � log3 (y � 4) � log3 13 11.
12. Use log2 3 � 1.5850 and log2 7 � 2.8074 to approximate 12.the value of log2 84.
Part II
Part I
NAME DATE PERIOD
SCORE 1010
Ass
essm
ent
y
xO
© Glencoe/McGraw-Hill 626 Glencoe Algebra 2
Chapter 10 Cumulative Review (Chapters 1–10)
NAME DATE PERIOD
1010
1. Name the sets of numbers to which ��16� belongs. (Lesson 1–2)
For Questions 2 and 3, use the following information.A new camp site will contain t tent sites, with 25 square meters of land each, and r recreational vehicle (RV) sites with 40 square meters of land each. No more than 90 camp sites can be built on the 3000 square meters of land available.
2. Write a system of inequalities to represent the number of sites built. Then list the coordinates of the vertices of the feasible region. (Lesson 3–4)
3. The site owner will charge $14 per tent site and $20 per RV site per day. Write a function for the total profit per day. Then determine the number of each type of site needed to earn a maximum profit, and find the maximum profit per day. (Lesson 3–4)
4. Simplify �3
27x3y�. (Lesson 5–6)
5. Solve 2x2 � x � 1 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. (Lesson 6–2)
6. Find all the zeros of the function h(x) � x3 � 5x2 � 8x � 6.(Lesson 7–5)
7. Find (f � g)(x), (f � g)(x), (f g)(x), and ��gf��(x) for
f(x) � x2 � 2x � 15 and g(x) � 2x � 1. (Lesson 7–7)
8. Write an equation for the hyperbola with vertices (0, 5) and (0, �5) and a conjugate axis at length 6 units. (Lesson 8–5)
9. Find the exact solution(s) of the system of equations x2 � y2 � 13x2 � 8y2 � 4. (Lesson 8–7)
For Questions 10 and 11, simplify.
10. �10
sm2n4
� � ��5ms3
n��
2(Lesson 9–1)
11. �x �
45
� � �5 �
3x
� (Lesson 9–2)
12. Identify the function represented by the equation y � �2x . (Lesson 9–5)
13. Write an exponential function whose graph passes through the points (0, 4) and (�2, 100). (Lesson 10–1)
14. Solve log�19
�x � �2. (Lesson 10–2)
15. Solve 65n � 542n � 3. Round to four decimal places. (Lesson 10–4)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
y
xO
Ass
essm
ent
Standardized Test Practice (Chapters 1–10)
© Glencoe/McGraw-Hill 627 Glencoe Algebra 2
NAME DATE PERIOD
1010
1. If �8x
� � x, which could be a value for x?
A. �1 B. 0 C. 2 D. �14
� 1.
2. If 0 � a � 1, which of the following increases as a decreases?
E. a � 1 F. a2 � 1 G. �a1
� H. a2 2.
3. If 3x � 2 is an odd integer, what is the next consecutive odd integer?A. 3x � 1 B. 3x � 3 C. 3x � 1 D. 3x 3.
4. Jody sold 4 more than twice the number of cars that Laura sold.If Laura sold c cars, how many more did Jody sell than Laura?E. 4 F. c � 4 G. 3c � 4 H. 2c � 4 4.
5. If 8 � 3z � 16 � 5z, then what is the value of 4z?A. �16 B. �4 C. 1 D. 12 5.
6. The radius of a wheel is 6 inches. How many revolutions will it make if it is rolled a distance of 288� inches?E. 8 F. 8� G. 24 H. 24� 6.
7. What is the 8th term in the sequence 3, 2, 0, �4, �12, …?A. �124 B. �60 C. �36 D. �144 7.
8. Which Venn diagram models the relationships among the sets A � {1, 2, 3}, B � {�4, 0}, and C � {positive integers}?E. F. G. H. 8.
9. A total of $270 is to be divided among four children. Each will receive an amount that is proportional to his or her age. If the children are 5, 10, 14, and 16 years old, how much money does the youngest child receive?A. $96 B. $6 C. $30 D. $54 9. DCBA
HGFE
B
A
CBA
C
BA
C
BA
C
DCBA
HGFE
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HGFE
DCBA
HGFE
DCBA
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
© Glencoe/McGraw-Hill 628 Glencoe Algebra 2
Standardized Test Practice (continued)
NAME DATE PERIOD
1010
NAME DATE PERIOD
10. If m2 � n2 � 140 and mn � 49, what is the value of (m � n)2?
11. What is the slope of a line that is perpendicular to the graph of 5x � 4y � 7?
12. If � � m in the figure shown,what is the value of d?
13. Find the perimeter of square EFGH if the areas of rectangle ABCD and square EFGH are equal.
Column A Column B
14. 14.
15. A right triangle has sides 6, 6, and c. 15.
16. �x32� � y 16.
17. 17.
18. S � {19, 22, 11, 17, 35} 18.
The mean of set SThe median of set S
DCBA
DCBA(�3)333(�3)104
yx
DCBA
c8
DCBA
DCBA14% of 230023% of 1400
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
10. 11.
12. 13.
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Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
d˚60˚
132˚m
�
A B
D
E F
H G
C
x � 18
2x
x
A
D
C
B
Unit 3 Test (Chapters 8–10)
© Glencoe/McGraw-Hill 629 Glencoe Algebra 2
1. Find the midpoint of the line segment with end points at 1.(�6, 2) and (3, 8).
2. Write an equation for the parabola with focus (4, 0) and 2.directrix y � 2.
3. Graph 16x2 � 9y2 � 144. 3.
4. Graph �x9
2� � �
(y �
92)2� � 1. 4.
5. Write an equation for the circle with center (�4, �7�) and 5.radius 5 units.
6. Write an equation for the ellipse with end points of the 6.major axis at (7, 1) and (�7, 1) and end points of the minor axis at (0, 5) and (0, �3).
7. Write 36x2 � 360x � 25y2 � 100y � 100 in standard form. 7.Then state whether the graph of the equation is a parabola,circle, ellipse, or hyperbola.
8. Find the exact solutions of the system of equations 8.x2 � 2y2 � 18 and x � 2y.
9. Simplify �4zx
2y3� � ��8z
x3
2y��2
. 9.
10. Simplify �d2d� 9� � �2d
5� 6�. 10.
11. Find the LCM of m2 � 4m � 5 and m2 � 8m � 7. 11.
12. Determine the equations of any vertical asymptotes and the 12.
value of x for any holes in the graph of f(x) ��x2 �
x1�
1x2� 18
�.
y
xO
y
xO
NAME DATE PERIOD
SCORE
Ass
essm
ent
© Glencoe/McGraw-Hill 630 Glencoe Algebra 2
Unit 3 Test (continued)(Chapters 8–10)
13. If y varies jointly as x and z and y � 100 when x � 10 and 13.z � 5, find y when x � 12 and z � 6.
14. Identify the type of function represented 14.by the graph.
15. Identify the function represented by y � �3x
�. 15.
16. Solve �t �
85
� � �tt
�
�
35
� � �13
�. 16.
17. Sketch the graph of y � 1.5(2)x. Then state the function’s 17.domain and range.
18. Determine whether y � 1.5��16��x
represents exponential 18.growth or decay.
19. Simplify x5 x�. 19.
For Questions 20–24, solve each equation or inequality.Round to four decimal places if necessary.
20. ��15
��t�2� 125 21. log4 (x � 9) � 2
22. log4 z � log4 (z � 3) � 1
23. 3.9m�4 � 10.21 24. e3x � 21
25. Evaluate log6 69. 25.
26. Evaluate ln e�3x. 26.
27. Use log5 2 � 0.4307 and log5 3 � 0.6826 to approximate the 27.value of log5 12.
28. Express log6 19 in terms of common logarithms. Then 28.approximate its value to four decimal places.
29. In a certain area, the sale price of new single-family homes 29.has increased 4.1% per year since 1992. If a house was purchased in this area in 1992 for $75,000 and the growth continues, what will the sale price be in 2006? Use y � a(1 � r)t and round to the nearest cent.
y
xO
NAME DATE PERIOD
20.
21.
22.
23.
24.
y
xO
Standardized Test PracticeStudent Record Sheet (Use with pages 572–573 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
1010
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 9
2 5 8 10
3 6
Solve the problem and write your answer in the blank.
For Questions 12–18, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
11 13 15 17
12 14 16 18
Select the best answer from the choices given and fill in the corresponding oval.
19 21 23
20 22 DCBADCBA
DCBADCBADCBA
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DCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 10-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Exp
on
enti
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill57
3G
lenc
oe A
lgeb
ra 2
Lesson 10-1
Exp
on
enti
al F
un
ctio
ns
An
exp
onen
tial
fu
nct
ion
has
th
e fo
rm y
�ab
x ,w
her
e a
�0,
b�
0,an
d b
�1.
1.T
he f
unct
ion
is c
ontin
uous
and
one
-to-
one.
Pro
per
ties
of
an2.
The
dom
ain
is t
he s
et o
f al
l rea
l num
bers
.
Exp
on
enti
al F
un
ctio
n3.
The
x-a
xis
is t
he a
sym
ptot
e of
the
gra
ph.
4.T
he r
ange
is t
he s
et o
f al
l pos
itive
num
bers
if a
�0
and
all n
egat
ive
num
bers
if a
�0.
5.T
he g
raph
con
tain
s th
e po
int
(0,
a).
Exp
on
enti
al G
row
thIf
a�
0 an
d b
�1,
the
fun
ctio
n y
�ab
xre
pres
ents
exp
onen
tial g
row
th.
and
Dec
ayIf
a�
0 an
d 0
�b
�1,
the
fun
ctio
n y
�ab
xre
pres
ents
exp
onen
tial d
ecay
.
Sk
etch
th
e gr
aph
of
y�
0.1(
4)x .
Th
en s
tate
th
e
fun
ctio
n’s
dom
ain
an
d r
ange
.M
ake
a ta
ble
of v
alu
es.C
onn
ect
the
poin
ts t
o fo
rm a
sm
ooth
cu
rve.
Th
e do
mai
n o
f th
e fu
nct
ion
is
all
real
nu
mbe
rs,w
hil
e th
e ra
nge
is
the
set
of a
ll p
osit
ive
real
nu
mbe
rs.
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
g
row
thor
dec
ay.
a.y
�0.
5(2)
xb
.y�
�2.
8(2)
xc.
y�
1.1(
0.5)
x
expo
nen
tial
gro
wth
,n
eith
er,s
ince
�2.
8,ex
pon
enti
al d
ecay
,sin
cesi
nce
th
e ba
se,2
,is
the
valu
e of
ais
les
s th
e ba
se,0
.5,i
s be
twee
ngr
eate
r th
an 1
than
0.
0 an
d 1
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
�3(
2)x
2.y
��
2 ��x
3.y
�0.
25(5
)x
Do
mai
n:
all r
eal
Do
mai
n:
all r
eal
Do
mai
n:
all r
eal
nu
mb
ers;
Ran
ge:
all
nu
mb
ers;
Ran
ge:
all
nu
mb
ers;
Ran
ge:
all
po
siti
ve r
eal n
um
ber
sn
egat
ive
real
nu
mb
ers
po
sitiv
e re
al n
um
ber
sD
eter
min
e w
het
her
eac
h f
un
ctio
n r
epre
sen
ts e
xpon
enti
al g
row
th o
r d
ecay
.
4.y
�0.
3(1.
2)x
gro
wth
5.y
��
5 ��x
nei
ther
6.y
�3(
10)�
xd
ecay
4 � 5
x
y
O
x
y
O
x
y
O
1 � 4
x�
10
12
3
y0.
025
0.1
0.4
1.6
6.4
x
y
O
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill57
4G
lenc
oe A
lgeb
ra 2
Exp
on
enti
al E
qu
atio
ns
and
Ineq
ual
itie
sA
ll t
he
prop
erti
es o
f ra
tion
al e
xpon
ents
that
you
kn
ow a
lso
appl
y to
rea
l ex
pon
ents
.Rem
embe
r th
at a
m�
an�
am
n,(
am)n
�am
n,
and
am
an�
am�
n.
Pro
per
ty o
f E
qu
alit
y fo
rIf
bis
a p
ositi
ve n
umbe
r ot
her
than
1,
Exp
on
enti
al F
un
ctio
ns
then
bx
�b
yif
and
only
if x
�y.
Pro
per
ty o
f In
equ
alit
y fo
rIf
b�
1
Exp
on
enti
al F
un
ctio
ns
then
bx
�b
yif
and
only
if x
�y
and
bx
�b
yif
and
only
if x
�y.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Exp
on
enti
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
Sol
ve 4
x�
1�
2x
�5 .
4x�
1�
2x
5O
rigin
al e
quat
ion
(22 )
x�
1�
2x
5R
ewrit
e 4
as 2
2 .
2(x
�1)
�x
5
Pro
p. o
f In
equa
lity
for
Exp
onen
tial
Fun
ctio
ns
2x�
2 �
x
5D
istr
ibut
ive
Pro
pert
y
x�
7S
ubtr
act
xan
d ad
d 2
to e
ach
side
.
Sol
ve 5
2x�
1�
.
52x
�1
�O
rigin
al in
equa
lity
52x
�1
�5�
3R
ewrit
e as
5�
3 .
2x�
1 �
�3
Pro
p. o
f In
equa
lity
for
Exp
onen
tial F
unct
ions
2x�
�2
Add
1 t
o ea
ch s
ide.
x�
�1
Div
ide
each
sid
e by
2.
Th
e so
luti
on s
et i
s {x
|x�
�1}
.
1� 12
5
1� 12
5
1� 12
5Ex
ampl
e1Ex
ampl
e1Ex
ampl
e2Ex
ampl
e2
Exer
cises
Exer
cises
Sim
pli
fy e
ach
exp
ress
ion
.
1.(3
�2� )�
2�2.
25�
2��
125�
2�3.
(x�2� y
3�2� )�
2�
955
�2�
or
3125
�2�
x2 y
6
4.(x�
6� )(x�
5� )5.
(x�6� )�
5�6.
(2x�
)(5x3
�)
x�6�
��
5�x�
30�10
x4�
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
on.
7.32
x�
1�
3x
23
8.23
x�
4x
24
9.32
x�
1�
�
10.4
x
1�
82x
3
�11
.8x
�2
�12
.252
x�
125x
2
6
13.4
�x�
�16
�5�
2014
.x�
3��
36���3 4�
615
.x�
2��
81� �1 8��
3
16.3
x�
4�
x�
117
.42x
�2
�2x
1
x�
18.5
2x�
125x
�5
x�
15
19.1
04x
1
�10
0x�
220
.73x
�49
x221
.82x
�5
�4x
8
x�
�x
�o
r x
�0
x�
�3 41 �3 � 2
5 � 2
5 � 31 � 27
2 � 31 � 16
7 � 4
1 � 21 � 9
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
A
Answers (Lesson 10-1)
Skil
ls P
ract
ice
Exp
on
enti
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill57
5G
lenc
oe A
lgeb
ra 2
Lesson 10-1
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
�3(
2)x
2.y
�2 �
�x
do
mai
n:
all r
eal n
um
ber
s;d
om
ain
:al
l rea
l nu
mb
ers;
ran
ge:
all p
osi
tive
nu
mb
ers
ran
ge:
all p
osi
tive
nu
mb
ers
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
gro
wth
or d
eca
y.
3.y
�3(
6)x
gro
wth
4.y
�2 �
�xd
ecay
5.y
�10
�x
dec
ay6.
y�
2(2.
5)x
gro
wth
Wri
te a
n e
xpon
enti
al f
un
ctio
n w
hos
e gr
aph
pas
ses
thro
ugh
th
e gi
ven
poi
nts
.
7.(0
,1)
and
(�1,
3)y
��
�x8.
(0,4
) an
d (1
,12)
y�
4(3)
x
9.(0
,3)
and
(�1,
6)y
�3 �
�x10
.(0,
5) a
nd
(1,1
5)y
�5(
3)x
11.(
0,0.
1) a
nd
(1,0
.5)
y�
0.1(
5)x
12.(
0,0.
2) a
nd
(1,1
.6)
y�
0.2(
8)x
Sim
pli
fy e
ach
exp
ress
ion
.
13. (3
�3� )�
3�27
14.(
x�2� )�
7�x�
14�
15.5
2�3�
�54
�3�
56�
3�16
.x3�
x�
x2�
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
on.
17.3
x�
9x
�2
18.2
2x
3�
321
19.4
9x�
x
�20
.43x
�2
�16
21.3
2x
5�
27x
522
.27x
�32
x
334 � 3
1 � 21 � 7
1 � 2
1 � 3
9 � 10
x
y
Ox
y
O
1 � 2
©G
lenc
oe/M
cGra
w-H
ill57
6G
lenc
oe A
lgeb
ra 2
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
�1.
5(2)
x2.
y�
4(3)
x3.
y�
3(0.
5)x
do
mai
n:
all r
eal
do
mai
n:
all r
eal
do
mai
n:
all r
eal
nu
mb
ers;
ran
ge:
all
nu
mb
ers;
ran
ge:
all
nu
mb
ers;
ran
ge:
all
po
siti
ve n
um
ber
sp
osi
tive
nu
mb
ers
po
siti
ve n
um
ber
s
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
gro
wth
or d
eca
y.
4.y
�5(
0.6)
xd
ecay
5.y
�0.
1(2)
xg
row
th6.
y�
5 �
4�x
dec
ay
Wri
te a
n e
xpon
enti
al f
un
ctio
n w
hos
e gr
aph
pas
ses
thro
ugh
th
e gi
ven
poi
nts
.
7.(0
,1)
and
(�1,
4)8.
(0,2
) an
d (1
,10)
9.(0
,�3)
an
d (1
,�1.
5)
y�
��x
y�
2(5)
xy
��
3(0.
5)x
10.(
0,0.
8) a
nd
(1,1
.6)
11.(
0,�
0.4)
an
d (2
,�10
)12
.(0,
�)
and
(3,8
�)
y�
0.8(
2)x
y�
�0.
4(5)
xy
��
(2)x
Sim
pli
fy e
ach
exp
ress
ion
.
13.(2
�2� )�
8�16
14.(
n�
3� )�75�
n15
15.y
�6�
�y5
�6�
y6�
6�
16.1
3�6�
�13
�24�
133�
6�17
.n3
n
�n
3 �
�18
.125
�11�
5�
11�52
�11�
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
on.
19.3
3x �
5�
81x
�3
20.7
6x�
72x
�20
�5
21.3
6n�
5�
94n
�3
n�
22.9
2x�
1�
27x
4
1423
.23n
�1
�
�nn
24
.164
n�
1�
1282
n
1
BIO
LOG
YF
or E
xerc
ises
25
and
26,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
init
ial
nu
mbe
r of
bac
teri
a in
a c
ult
ure
is
12,0
00.T
he
nu
mbe
r af
ter
3 da
ys i
s 96
,000
.
25.W
rite
an
exp
onen
tial
fu
nct
ion
to
mod
el t
he
popu
lati
on y
of b
acte
ria
afte
r x
days
.y
�12
,000
(2)x
26.H
ow m
any
bact
eria
are
th
ere
afte
r 6
days
?76
8,00
0
27.E
DU
CA
TIO
NA
col
lege
wit
h a
gra
duat
ing
clas
s of
400
0 st
ude
nts
in
th
e ye
ar 2
002
pred
icts
th
at i
t w
ill
hav
e a
grad
uat
ing
clas
s of
486
2 in
4 y
ears
.Wri
te a
n e
xpon
enti
alfu
nct
ion
to
mod
el t
he
nu
mbe
r of
stu
den
ts y
in t
he
grad
uat
ing
clas
s t
year
s af
ter
2002
.y
�40
00(1
.05)
t
11 � 21 � 6
1 � 8
1 � 2
1 � 4
x
y
Ox
y
Ox
y
OPra
ctic
e (
Ave
rag
e)
Exp
on
enti
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 10-1)
Readin
g t
o L
earn
Math
em
ati
csE
xpo
nen
tial
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill57
7G
lenc
oe A
lgeb
ra 2
Lesson 10-1
Pre-
Act
ivit
yH
ow d
oes
an e
xpon
enti
al f
un
ctio
n d
escr
ibe
tou
rnam
ent
pla
y?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-1 a
t th
e to
p of
pag
e 52
3 in
you
r te
xtbo
ok.
How
man
y ro
un
ds o
f pl
ay w
ould
be
nee
ded
for
a to
urn
amen
t w
ith
100
play
ers?
7
Rea
din
g t
he
Less
on
1.In
dica
te w
het
her
eac
h o
f th
e fo
llow
ing
stat
emen
ts a
bou
t th
e ex
pon
enti
al f
un
ctio
n
y�
10x
is t
rue
or f
alse
.
a.T
he
dom
ain
is
the
set
of a
ll p
osit
ive
real
nu
mbe
rs.
fals
e
b.
Th
e y-
inte
rcep
t is
1.
tru
e
c.T
he
fun
ctio
n i
s on
e-to
-on
e.tr
ue
d.
Th
e y-
axis
is
an a
sym
ptot
e of
th
e gr
aph
.fa
lse
e.T
he
ran
ge i
s th
e se
t of
all
rea
l n
um
bers
.fa
lse
2.D
eter
min
e w
het
her
eac
h f
un
ctio
n r
epre
sen
ts e
xpon
enti
al g
row
thor
dec
ay.
a.y
�0.
2(3)
x .g
row
thb
.y�
3 ��x .
dec
ayc.
y�
0.4(
1.01
)x.
gro
wth
3.S
upp
ly t
he
reas
on f
or e
ach
ste
p in
th
e fo
llow
ing
solu
tion
of
an e
xpon
enti
al e
quat
ion
.
92x
�1
�27
xO
rigi
nal
equ
atio
n
(32 )
2x�
1�
(33 )
xR
ewri
te e
ach
sid
e w
ith
a b
ase
of
3.32
(2x
�1)
�33
xP
ow
er o
f a
Po
wer
2(2x
�1)
�3x
Pro
per
ty o
f E
qu
alit
y fo
r E
xpo
nen
tial
Fu
nct
ion
s4x
�2
�3x
Dis
trib
uti
ve P
rop
erty
x�
2 �
0S
ub
trac
t 3x
fro
m e
ach
sid
e.x
�2
Ad
d 2
to
eac
h s
ide.
Hel
pin
g Y
ou
Rem
emb
er
4.O
ne
way
to
rem
embe
r th
at p
olyn
omia
l fu
nct
ion
s an
d ex
pon
enti
al f
un
ctio
ns
are
diff
eren
tis
to
con
tras
t th
e po
lyn
omia
l fu
nct
ion
y�
x2an
d th
e ex
pon
enti
al f
un
ctio
n y
�2x
.Tel
l at
leas
t th
ree
way
s th
ey a
re d
iffe
ren
t.
Sam
ple
an
swer
:In
y�
x2 ,
the
vari
able
xis
a b
ase,
but
in y
�2x
,th
eva
riab
le x
is a
n e
xpo
nen
t.T
he
gra
ph
of
y�
x2
is s
ymm
etri
c w
ith
res
pec
tto
th
e y-
axis
,bu
t th
e g
rap
h o
f y
�2x
is n
ot.
Th
e g
rap
h o
f y
�x
2to
uch
esth
e x-
axis
at
(0,0
),bu
t th
e g
rap
h o
f y
�2x
has
th
e x-
axis
as
an a
sym
pto
te.
You
can
co
mp
ute
th
e va
lue
of
y�
x2
men
tally
fo
r x
�10
0,bu
t yo
u c
ann
ot
com
pu
te t
he
valu
e o
f y
�2x
men
tally
fo
r x
�10
0.
2 � 5
©G
lenc
oe/M
cGra
w-H
ill57
8G
lenc
oe A
lgeb
ra 2
Fin
din
g S
olu
tio
ns
of
xy�
yx
Per
hap
s yo
u h
ave
not
iced
th
at i
f x
and
yar
e in
terc
han
ged
in e
quat
ion
s su
chas
x�
yan
d xy
�1,
the
resu
ltin
g eq
uat
ion
is
equ
ival
ent
to t
he
orig
inal
equ
atio
n.T
he
sam
e is
tru
e of
th
e eq
uat
ion
xy
�yx
.How
ever
,fin
din
gso
luti
ons
of x
y�
yxan
d dr
awin
g it
s gr
aph
is
not
a s
impl
e pr
oces
s.
Sol
ve e
ach
pro
ble
m.A
ssu
me
that
xan
d y
are
pos
itiv
e re
al n
um
ber
s.
1.If
a�
0,w
ill
(a,a
) be
a s
olu
tion
of
xy�
yx?
Just
ify
you
r an
swer
.
Yes,
sin
ce a
a�
aam
ust
be
tru
e (R
efle
xive
Pro
p.o
f E
qu
alit
y).
2.If
c�
0,d
�0,
and
(c,d
) is
a s
olu
tion
of
xy�
yx,w
ill
(d,c
) al
so
be a
sol
uti
on?
Just
ify
you
r an
swer
.
Yes;
rep
laci
ng
xw
ith
d,y
wit
h c
giv
es d
c�
cd;
but
if (
c,d
) is
a s
olu
tio
n,
cd
�d
c .S
o,b
y th
e S
ymm
etri
c P
rop
erty
of
Eq
ual
ity,
dc
�c
dis
tru
e.
3.U
se 2
as
a va
lue
for
yin
xy
�yx
.Th
e eq
uat
ion
bec
omes
x2
�2x
.
a.F
ind
equ
atio
ns
for
two
fun
ctio
ns,
f(x)
an
d g(
x) t
hat
you
cou
ld g
raph
to
fin
d th
e so
luti
ons
of x
2�
2x.T
hen
gra
ph t
he
fun
ctio
ns
on a
sep
arat
esh
eet
of g
raph
pap
er.
f(x)
�x
2 ,g
(x)
�2
x
See
stu
den
ts’g
rap
hs.
b.
Use
th
e gr
aph
you
dre
w f
or p
art
ato
sta
te t
wo
solu
tion
s fo
r x2
�2x
.T
hen
use
th
ese
solu
tion
s to
sta
te t
wo
solu
tion
s fo
r xy
�yx
.2,
4;(2
,2),
(4,2
)
4.In
th
is e
xerc
ise,
a gr
aph
ing
calc
ula
tor
wil
l be
ver
y h
elpf
ul.
Use
th
e te
chn
iqu
e of
Exe
rcis
e 3
to c
ompl
ete
the
tabl
es b
elow
.Th
en g
raph
xy
�yx
for
posi
tive
val
ues
of
xan
d y.
If t
her
e ar
e as
ympt
otes
,sh
ow t
hem
in
you
rdi
agra
m u
sin
g do
tted
lin
es.N
ote
that
in
th
e ta
ble,
som
e va
lues
of
yca
llfo
r on
e va
lue
of x
,oth
ers
call
for
tw
o.
x
y
O
xy
44
24
55
1.8
5
88
1.5
8
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1 x
y
�1 2��1 2�
�3 4��3 4�
11
22
42
33
2.5
3
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
A
Answers (Lesson 10-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Lo
gar
ith
ms
and
Lo
gar
ith
mic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill57
9G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Log
arit
hm
ic F
un
ctio
ns
and
Exp
ress
ion
s
Def
init
ion
of
Lo
gar
ith
m
Let
ban
d x
be p
ositi
ve n
umbe
rs,
b�
1. T
he lo
garit
hm o
f x
with
bas
e b
is d
enot
ed
wit
h B
ase
blo
g bx
and
is d
efin
ed a
s th
e ex
pone
nt y
that
mak
es t
he e
quat
ion
by
�x
true
.
Th
e in
vers
e of
th
e ex
pon
enti
al f
un
ctio
n y
�bx
is t
he
loga
rith
mic
fu
nct
ion
x�
by.
Th
is f
un
ctio
n i
s u
sual
ly w
ritt
en a
s y
�lo
g bx.
1.T
he f
unct
ion
is c
ontin
uous
and
one
-to-
one.
Pro
per
ties
of
2.T
he d
omai
n is
the
set
of
all p
ositi
ve r
eal n
umbe
rs.
Lo
gar
ith
mic
Fu
nct
ion
s3.
The
y-a
xis
is a
n as
ympt
ote
of t
he g
raph
.4.
The
ran
ge is
the
set
of
all r
eal n
umbe
rs.
5.T
he g
raph
con
tain
s th
e po
int
(0,
1).
Wri
te a
n e
xpon
enti
al e
qu
atio
n e
qu
ival
ent
to l
og3
243
�5.
35�
243
Wri
te a
log
arit
hm
ic e
qu
atio
n e
qu
ival
ent
to 6
�3
�.
log 6
��
3
Eva
luat
e lo
g 816
.
8�4 3�
�16
,so
log 8
16 �
.
Wri
te e
ach
eq
uat
ion
in
log
arit
hm
ic f
orm
.
1.27
�12
82.
3�4
�3.
��3
�
log
212
8 �
7lo
g3
��
4lo
g�1 7�
�3
Wri
te e
ach
eq
uat
ion
in
exp
onen
tial
for
m.
4.lo
g 15
225
�2
5.lo
g 3�
�3
6.lo
g 432
�
152
�22
53�
3�
4�5 2��
32
Eva
luat
e ea
ch e
xpre
ssio
n.
7.lo
g 464
38.
log 2
646
9.lo
g 100
100,
000
2.5
10.l
og5
625
411
.log
2781
12.l
og25
5
13.l
og2
�7
14.l
og10
0.00
001
�5
15.l
og4
�2.
51 � 32
1� 12
8
1 � 24 � 31 � 27
5 � 21 � 27
1� 34
31 � 81
1� 34
31 � 7
1 � 81
4 � 3
1� 21
6
1� 21
6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill58
0G
lenc
oe A
lgeb
ra 2
Solv
e Lo
gar
ith
mic
Eq
uat
ion
s an
d In
equ
alit
ies
Lo
gar
ith
mic
to
If
b�
1, x
�0,
and
log b
x�
y, t
hen
x�
by .
Exp
on
enti
al In
equ
alit
yIf
b�
1, x
�0,
and
log b
x�
y, t
hen
0 �
x�
by.
Pro
per
ty o
f E
qu
alit
y fo
r If
bis
a p
ositi
ve n
umbe
r ot
her
than
1,
Lo
gar
ith
mic
Fu
nct
ion
sth
en lo
g bx
�lo
g by
if an
d on
ly if
x�
y.
Pro
per
ty o
f In
equ
alit
y fo
r If
b�
1, t
hen
log b
x�
log b
yif
and
only
if x
�y,
L
og
arit
hm
ic F
un
ctio
ns
and
log b
x�
log b
yif
and
only
if x
�y.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Lo
gar
ith
ms
and
Lo
gar
ith
mic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
Sol
ve l
og2
2x�
3.lo
g 22x
�3
Orig
inal
equ
atio
n
2x�
23D
efin
ition
of
loga
rithm
2x�
8S
impl
ify.
x�
4S
impl
ify.
Th
e so
luti
on i
s x
�4.
Sol
ve l
og5
(4x
�3)
�3.
log 5
(4x
�3)
�3
Orig
inal
equ
atio
n
0 �
4x�
3 �
53Lo
garit
hmic
to
expo
nent
ial i
nequ
ality
3 �
4x�
125
3
Add
ition
Pro
pert
y of
Ine
qual
ities
�x
�32
Sim
plify
.
Th
e so
luti
on s
et i
s �x |
�x
�32
.3 � 4
3 � 4
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
1.lo
g 232
�3x
2.lo
g 32c
��
2
3.lo
g 2x
16 �
�2
4.lo
g 25�
��10
5.lo
g 4(5
x
1) �
23
6.lo
g 8(x
�5)
�9
7.lo
g 4(3
x�
1) �
log 4
(2x
3)
48.
log 2
(x2
�6)
�lo
g 2(2
x
2)4
9.lo
g x
427
�3
�1
10.l
og2
(x
3) �
413
11.l
ogx
1000
�3
1012
.log
8(4
x
4) �
215
13.l
og2
2x�
2x
�2
14.l
og5
x�
2x
�25
15.l
og2
(3x
1)
�4
��
x�
516
.log
4(2
x) �
�x
�
17.l
og3
(x
3) �
3�
3 �
x�
2418
.log
276x
�x
�3 � 2
2 � 3
1 � 41 � 2
1 � 3
2 � 3
1 � 2x � 2
1 � 8
1 � 185 � 3
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 10-2)
Skil
ls P
ract
ice
Lo
gar
ith
ms
and
Lo
gar
ith
mic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill58
1G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Wri
te e
ach
eq
uat
ion
in
log
arit
hm
ic f
orm
.
1.23
�8
log
28
�3
2.32
�9
log
39
�2
3.8�
2�
log
8�
�2
4.�
�2�
log
�1 3��
2
Wri
te e
ach
eq
uat
ion
in
exp
onen
tial
for
m.
5.lo
g 324
3 �
535
�24
36.
log 4
64 �
343
�64
7.lo
g 93
�9�1 2�
�3
8.lo
g 5�
�2
5�2
�
Eva
luat
e ea
ch e
xpre
ssio
n.
9.lo
g 525
210
.log
93
11.l
og10
1000
312
.log
125
5
13.l
og4
�3
14.l
og5
�4
15.l
og8
833
16.l
og27
�
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
ons.
17.l
og3
x�
524
318
.log
2x
�3
8
19.l
og4
y�
00
�y
�1
20.l
og�1 4�
x�
3
21.l
og2
n�
�2
n�
22.l
ogb
3 �
9
23.l
og6
(4x
12
) �
26
24.l
og2
(4x
�4)
�5
x�
9
25.l
og3
(x
2) �
log 3
(3x)
126
.log
6(3
y�
5)
log 6
(2y
3)
y
8
1 � 21 � 4
1 � 641 � 31 � 31
� 625
1 � 64
1 � 3
1 � 2
1 � 251 � 25
1 � 2
1 � 91 � 9
1 � 31 � 64
1 � 64
©G
lenc
oe/M
cGra
w-H
ill58
2G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
eq
uat
ion
in
log
arit
hm
ic f
orm
.
1.53
�12
5lo
g5
125
�3
2.70
�1
log
71
�0
3.34
�81
log
381
�4
4.3�
4�
5.�
�3�
6.77
76�1 5�
�6
log
3�
�4
log
�1 4��
3lo
g77
766
�
Wri
te e
ach
eq
uat
ion
in
exp
onen
tial
for
m.
7.lo
g 621
6 �
363
�21
68.
log 2
64 �
626
�64
9.lo
g 3�
�4
3�4
�
10.l
og10
0.00
001
��
511
.log
255
�12
.log
328
�
10�
5�
0.00
001
25�1 2�
�5
32�3 5�
�8
Eva
luat
e ea
ch e
xpre
ssio
n.
13.l
og3
814
14.l
og10
0.00
01�
415
.log
2�
416
.log
�1 3�27
�3
17.l
og9
10
18.l
og8
419
.log
7�
220
.log
664
4
21.l
og3
�1
22.l
og4
�4
23.l
og9
9(n
1)
n�
124
.2lo
g 232
32
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
ons.
25.l
og10
n�
�3
26.l
og4
x�
3x
�64
27.l
og4
x�
8
28.l
og�1 5�
x�
�3
125
29.l
og7
q�
00
�q
�1
30.l
og6
(2y
8)
2
y
14
31.l
ogy
16 �
�4
32.l
ogn
��
32
33.l
ogb
1024
�5
4
34.l
og8
(3x
7)
�lo
g 8(7
x
4)35
.log
7(8
x
20)
�lo
g 7(x
6)
36.l
og3
(x2
�2)
�lo
g 3x
x�
�2
2
37.S
OU
ND
Sou
nds
th
at r
each
lev
els
of 1
30 d
ecib
els
or m
ore
are
pain
ful
to h
um
ans.
Wh
atis
th
e re
lati
ve i
nte
nsi
ty o
f 13
0 de
cibe
ls?
1013
38.I
NV
ESTI
NG
Mar
ia i
nve
sts
$100
0 in
a s
avin
gs a
ccou
nt
that
pay
s 8%
in
tere
stco
mpo
un
ded
ann
ual
ly.T
he
valu
e of
th
e ac
cou
nt
Aat
th
e en
d of
fiv
e ye
ars
can
be
dete
rmin
ed f
rom
th
e eq
uat
ion
log
A�
log[
1000
(1
0.08
)5].
Fin
d th
e va
lue
of A
to t
he
nea
rest
dol
lar.
$146
9
3 � 4
1 � 81 � 2
3 � 21
� 1000
1� 25
61 � 3
1 � 492 � 3
1 � 16
3 � 51 � 2
1 � 811 � 81
1 � 51 � 64
1 � 81
1 � 641 � 4
1 � 81Pra
ctic
e (
Ave
rag
e)
Lo
gar
ith
ms
and
Lo
gar
ith
mic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
A
Answers (Lesson 10-2)
Readin
g t
o L
earn
Math
em
ati
csL
og
arit
hm
s an
d L
og
arit
hm
ic F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill58
3G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Pre-
Act
ivit
yW
hy
is a
log
arit
hm
ic s
cale
use
d t
o m
easu
re s
oun
d?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-2 a
t th
e to
p of
pag
e 53
1 in
you
r te
xtbo
ok.
How
man
y ti
mes
lou
der
than
a w
his
per
is n
orm
al c
onve
rsat
ion
?10
4o
r 10
,000
tim
es
Rea
din
g t
he
Less
on
1.a.
Wri
te a
n e
xpon
enti
al e
quat
ion
th
at i
s eq
uiv
alen
t to
log
381
�4.
34�
81
b.
Wri
te a
log
arit
hm
ic e
quat
ion
th
at i
s eq
uiv
alen
t to
25�
�1 2��
.lo
g25
��
c.W
rite
an
exp
onen
tial
equ
atio
n t
hat
is
equ
ival
ent
to l
og4
1 �
0.40
�1
d.
Wri
te a
log
arit
hm
ic e
quat
ion
th
at i
s eq
uiv
alen
t to
10�
3�
0.00
1.lo
g10
0.00
1 �
�3
e.W
hat
is
the
inve
rse
of t
he
fun
ctio
n y
�5x
?y
�lo
g5
x
f.W
hat
is
the
inve
rse
of t
he
fun
ctio
n y
�lo
g 10
x?y
�10
x
2.M
atch
eac
h f
un
ctio
n w
ith
its
gra
ph.
a.y
�3x
IVb
.y�
log 3
xI
c.y
��
�xII
I.II
.II
I.
3.In
dica
te w
het
her
eac
h o
f th
e fo
llow
ing
stat
emen
ts a
bou
t th
e ex
pon
enti
al f
un
ctio
n
y�
log 5
xis
tru
eor
fal
se.
a.T
he
y-ax
is i
s an
asy
mpt
ote
of t
he
grap
h.
tru
eb
.T
he
dom
ain
is
the
set
of a
ll r
eal
nu
mbe
rs.
fals
ec.
Th
e gr
aph
con
tain
s th
e po
int
(5,0
).fa
lse
d.
Th
e ra
nge
is
the
set
of a
ll r
eal
nu
mbe
rs.
tru
ee.
Th
e y-
inte
rcep
t is
1.
fals
e
Hel
pin
g Y
ou
Rem
emb
er4.
An
im
port
ant
skil
l n
eede
d fo
r w
orki
ng
wit
h l
ogar
ith
ms
is c
han
gin
g an
equ
atio
n b
etw
een
loga
rith
mic
and
exp
onen
tial
for
ms.
Usi
ng t
he w
ords
bas
e,ex
pone
nt,a
nd l
ogar
ithm
,des
crib
ean
eas
y w
ay t
o re
mem
ber
and
appl
y th
e pa
rt o
f th
e de
fin
itio
n o
f lo
gari
thm
th
at s
ays,
“log
bx
�y
if a
nd
only
if
by
�x.
”S
amp
le a
nsw
er:
In t
hes
e eq
uat
ion
s,b
stan
ds
for
bas
e.In
log
fo
rm,b
is t
he
sub
scri
pt,
and
in e
xpo
nen
tial
fo
rm,b
is t
he
num
ber
th
at is
rai
sed
to
a p
ower
.A lo
gar
ithm
is a
n e
xpo
nen
t,so
y,w
hic
h is
the
log
in t
he
first
eq
uat
ion
,bec
om
es t
he
exp
on
ent
in t
he
seco
nd
eq
uat
ion
.
x
y
Ox
y
O
x
y
O
1 � 3
1 � 21 � 5
1 � 5
©G
lenc
oe/M
cGra
w-H
ill58
4G
lenc
oe A
lgeb
ra 2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
Mu
sica
l Rel
atio
nsh
ips
Th
e fr
equ
enci
es o
f n
otes
in
a m
usi
cal
scal
e th
at a
re o
ne
octa
ve a
part
are
rela
ted
by a
n e
xpon
enti
al e
quat
ion
.For
th
e ei
ght
C n
otes
on
a p
ian
o,th
eeq
uat
ion
is
Cn
�C
12n
�1 ,
wh
ere
Cn
repr
esen
ts t
he
freq
uen
cy o
f n
ote
Cn.
1.F
ind
the
rela
tion
ship
bet
wee
n C
1an
d C
2.C
2�
2C1
2.F
ind
the
rela
tion
ship
bet
wee
n C
1an
d C
4.C
4�
8C1
Th
e fr
equ
enci
es o
f co
nse
cuti
ve n
otes
are
rel
ated
by
a
com
mon
rat
io r
.Th
e ge
ner
al e
quat
ion
is
f n�
f 1rn
�1 .
3.If
th
e fr
equ
ency
of
mid
dle
C i
s 26
1.6
cycl
es p
er s
econ
d an
d th
e fr
equ
ency
of
the
nex
t h
igh
er C
is
523.
2 cy
cles
pe
r se
con
d,fi
nd
the
com
mon
rat
io r
.(H
int:
Th
e tw
o C
’s
are
12 n
otes
apa
rt.)
Wri
te t
he
answ
er a
s a
radi
cal
expr
essi
on.
r�
12 �2�
4.S
ubs
titu
te d
ecim
al v
alu
es f
or r
and
f 1to
fin
d a
spec
ific
eq
uat
ion
for
fn.
f n�
261.
1(1.
0594
6)n
�1
5.F
ind
the
freq
uen
cy o
f F
#ab
ove
mid
dle
C.
f 7�
261.
6(1.
0594
6)6
�36
9.95
6.F
rets
are
a s
erie
s of
rid
ges
plac
ed a
cros
s th
e fi
nge
rboa
rd o
f a
guit
ar.T
hey
are
spac
ed s
o th
at t
he
sou
nd
mad
e by
pre
ssin
g a
stri
ng
agai
nst
on
e fr
eth
as a
bou
t 1.
0595
tim
es t
he
wav
elen
gth
of
the
sou
nd
mad
e by
usi
ng
the
nex
t fr
et.T
he
gen
eral
equ
atio
n i
s w
n�
w0(
1.05
95)n
.Des
crib
e th
ear
ran
gem
ent
of t
he
fret
s on
a g
uit
ar.
Th
e fr
ets
are
spac
ed in
a
log
arit
hm
ic s
cale
.
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 10-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pro
per
ties
of
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
5G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Pro
per
ties
of
Log
arit
hm
sP
rope
rtie
s of
exp
onen
ts c
an b
e u
sed
to d
evel
op t
he
foll
owin
g pr
oper
ties
of
loga
rith
ms.
Pro
du
ct P
rop
erty
F
or a
ll po
sitiv
e nu
mbe
rs m
, n,
and
b,
whe
re b
�1,
o
f L
og
arit
hm
slo
g bm
n�
log b
m
log b
n.
Qu
oti
ent
Pro
per
ty
For
all
posi
tive
num
bers
m,
n, a
nd b
, w
here
b�
1,
of
Lo
gar
ith
ms
log b
�m n��
log b
m�
log b
n.
Po
wer
Pro
per
ty
For
any
rea
l num
ber
pan
d po
sitiv
e nu
mbe
rs m
and
b,
of
Lo
gar
ith
ms
whe
re b
�1,
log b
mp
�p
log b
m.
Use
log
328
3.
0331
an
d l
og3
4
1.26
19 t
o ap
pro
xim
ate
the
valu
e of
eac
h e
xpre
ssio
n.
Exam
ple
Exam
ple
a.lo
g 336
log 3
36�
log 3
(32
�4)
�lo
g 332
lo
g 34
�2
lo
g 34
2
1.
2619
3.
2619
b.
log 3
7
log 3
7�
log 3
��
�lo
g 328
�lo
g 34
3.
0331
�1.
2619
1.
7712
c.lo
g 325
6
log 3
256
�lo
g 3(4
4 )�
4 �
log 3
4
4(1.
2619
)
5.04
76
28 � 4
Exer
cises
Exer
cises
Use
log
123
0.
4421
an
d l
og12
7
0.78
31 t
o ev
alu
ate
each
exp
ress
ion
.
1.lo
g 12
211.
2252
2.lo
g 12
0.34
103.
log 1
249
1.56
62
4.lo
g 12
361.
4421
5.lo
g 12
631.
6673
6.lo
g 12
�0.
2399
7.lo
g 12
0.20
228.
log 1
216
,807
3.91
559.
log 1
244
12.
4504
Use
log
53
0.
6826
an
d l
og5
4
0.86
14 t
o ev
alu
ate
each
exp
ress
ion
.
10.l
og5
121.
5440
11.l
og5
100
2.86
1412
.log
50.
75�
0.17
88
13.l
og5
144
3.08
8014
.log
50.
3250
15.l
og5
375
3.68
26
16.l
og5
1.3�
0.17
8817
.log
5�
0.35
7618
.log
51.
7304
81 � 59 � 1627 � 16
81 � 49
27 � 49
7 � 3
©G
lenc
oe/M
cGra
w-H
ill58
6G
lenc
oe A
lgeb
ra 2
Solv
e Lo
gar
ith
mic
Eq
uat
ion
sYo
u c
an u
se t
he
prop
erti
es o
f lo
gari
thm
s to
sol
veeq
uat
ion
s in
volv
ing
loga
rith
ms.
Sol
ve e
ach
eq
uat
ion
.
a.2
log 3
x�
log 3
4 �
log 3
25
2 lo
g 3x
�lo
g 34
�lo
g 325
Orig
inal
equ
atio
n
log 3
x2�
log 3
4 �
log 3
25P
ower
Pro
pert
y
log 3
�lo
g 325
Quo
tient
Pro
pert
y
�25
Pro
pert
y of
Equ
ality
for
Log
arith
mic
Fun
ctio
ns
x2�
100
Mul
tiply
eac
h si
de b
y 4.
x�
�10
Take
the
squ
are
root
of
each
sid
e.
Sin
ce l
ogar
ith
ms
are
un
defi
ned
for
x�
0,�
10 i
s an
ext
ran
eou
s so
luti
on.
Th
e on
ly s
olu
tion
is
10.
b.
log 2
x
log 2
(x
2) �
3
log 2
x
log 2
(x
2) �
3O
rigin
al e
quat
ion
log 2
x(x
2)
�3
Pro
duct
Pro
pert
y
x(x
2)
�23
Def
initi
on o
f lo
garit
hm
x2
2x�
8D
istr
ibut
ive
Pro
pert
y
x2
2x �
8 �
0S
ubtr
act
8 fr
om e
ach
side
.
(x
4)(x
�2)
�0
Fac
tor.
x�
2or
x�
�4
Zer
o P
rodu
ct P
rope
rty
Sin
ce l
ogar
ith
ms
are
un
defi
ned
for
x�
0,�
4 is
an
ext
ran
eou
s so
luti
on.
Th
e on
ly s
olu
tion
is
2.
Sol
ve e
ach
eq
uat
ion
.Ch
eck
you
r so
luti
ons.
1.lo
g 54
lo
g 52x
�lo
g 524
32.
3 lo
g 46
�lo
g 48
�lo
g 4x
27
3.lo
g 625
lo
g 6x
�lo
g 620
44.
log 2
4 �
log 2
(x
3) �
log 2
8�
5.lo
g 62x
�lo
g 63
�lo
g 6(x
�1)
36.
2 lo
g 4(x
1)
�lo
g 4(1
1 �
x)2
7.lo
g 2x
�3
log 2
5 �
2 lo
g 210
12,5
008.
3 lo
g 2x
�2
log 2
5x�
210
0
9.lo
g 3(c
3)
�lo
g 3(4
c�
1) �
log 3
510
.log
5(x
3)
�lo
g 5(2
x�
1) �
24 � 7
8 � 19
5 � 21 � 2
x2� 4x2� 4
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Pro
per
ties
of
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
A
Answers (Lesson 10-3)
Skil
ls P
ract
ice
Pro
per
ties
of
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
7G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Use
log
23
1.
5850
an
d l
og2
5
2.32
19 t
o ap
pro
xim
ate
the
valu
e of
eac
hex
pre
ssio
n.
1.lo
g 225
4.64
382.
log 2
274.
755
3.lo
g 2�
0.73
694.
log 2
0.73
69
5.lo
g 215
3.90
696.
log 2
455.
4919
7.lo
g 275
6.22
888.
log 2
0.6
�0.
7369
9.lo
g 2�
1.58
5010
.log
20.
8481
Sol
ve e
ach
eq
uat
ion
.Ch
eck
you
r so
luti
ons.
11.l
og10
27 �
3 lo
g 10
x3
12.3
log
74
�2
log 7
b8
13.l
og4
5
log 4
x�
log 4
6012
14.l
og6
2c
log 6
8 �
log 6
805
15.l
og5
y�
log 5
8 �
log 5
18
16.l
og2
q�
log 2
3 �
log 2
721
17.l
og9
4
2 lo
g 95
�lo
g 9w
100
18.3
log
82
�lo
g 84
�lo
g 8b
2
19.l
og10
x
log 1
0(3
x�
5) �
log 1
02
220
.log
4x
lo
g 4(2
x�
3) �
log 4
22
21.l
og3
d
log 3
3 �
39
22.l
og10
y�
log 1
0(2
�y)
�0
1
23.l
og2
s
2 lo
g 25
�0
24.l
og2
(x
4) �
log 2
(x�
3) �
34
25.l
og4
(n
1) �
log 4
(n�
2) �
13
26.l
og5
10
log 5
12 �
3 lo
g 52
lo
g 5a
15
1 � 25
9 � 51 � 3
5 � 33 � 5
©G
lenc
oe/M
cGra
w-H
ill58
8G
lenc
oe A
lgeb
ra 2
Use
log
105
0.
6990
an
d l
og10
7
0.84
51 t
o ap
pro
xim
ate
the
valu
e of
eac
hex
pre
ssio
n.
1.lo
g 10
351.
5441
2.lo
g 10
251.
3980
3.lo
g 10
0.14
614.
log 1
0�
0.14
61
5.lo
g 10
245
2.38
926.
log 1
017
52.
2431
7.lo
g 10
0.2
�0.
6990
8.lo
g 10
0.55
29
Sol
ve e
ach
eq
uat
ion
.Ch
eck
you
r so
luti
ons.
9.lo
g 7n
�lo
g 78
410
.log
10u
�lo
g 10
48
11.l
og6
x
log 6
9 �
log 6
546
12.l
og8
48 �
log 8
w�
log 8
412
13.l
og9
(3u
14
) �
log 9
5 �
log 9
2u2
14.4
log
2x
lo
g 25
�lo
g 240
53
15.l
og3
y�
�lo
g 316
lo
g 364
16.l
og2
d�
5 lo
g 22
�lo
g 28
4
17.l
og10
(3m
�5)
lo
g 10
m�
log 1
02
218
.log
10(b
3)
lo
g 10
b�
log 1
04
1
19.l
og8
(t
10)
�lo
g 8(t
�1)
�lo
g 812
220
.log
3(a
3)
lo
g 3(a
2)
�lo
g 36
0
21.l
og10
(r
4) �
log 1
0r
�lo
g 10
(r
1)2
22.l
og4
(x2
�4)
�lo
g 4(x
2)
�lo
g 41
3
23.l
og10
4
log 1
0w
�2
2524
.log
8(n
�3)
lo
g 8(n
4)
�1
4
25.3
log
5(x
2
9) �
6 �
0�
426
.log
16(9
x
5) �
log 1
6(x
2�
1) �
3
27.l
og6
(2x
�5)
1
�lo
g 6(7
x
10)
828
.log
2(5
y
2) �
1 �
log 2
(1 �
2y)
0
29.l
og10
(c2
�1)
�2
�lo
g 10
(c
1)10
130
.log
7x
2
log 7
x�
log 7
3 �
log 7
726
31.S
OU
ND
Th
e lo
udn
ess
Lof
a s
oun
d in
dec
ibel
s is
giv
en b
y L
�10
log
10R
,wh
ere
Ris
th
eso
un
d’s
rela
tive
in
ten
sity
.If
the
inte
nsi
ty o
f a
cert
ain
sou
nd
is t
ripl
ed,b
y h
ow m
any
deci
bels
doe
s th
e so
un
d in
crea
se?
abo
ut
4.8
db
32.E
AR
THQ
UA
KES
An
ear
thqu
ake
rate
d at
3.5
on
th
e R
ich
ter
scal
e is
fel
t by
man
y pe
ople
,an
d an
ear
thqu
ake
rate
d at
4.5
may
cau
se l
ocal
dam
age.
Th
e R
ich
ter
scal
e m
agn
itu
dere
adin
g m
is g
iven
by
m�
log 1
0x,
wh
ere
xre
pres
ents
th
e am
plit
ude
of
the
seis
mic
wav
eca
usi
ng
grou
nd
mot
ion
.How
man
y ti
mes
gre
ater
is
the
ampl
itu
de o
f an
ear
thqu
ake
that
mea
sure
s 4.
5 on
th
e R
ich
ter
scal
e th
an o
ne
that
mea
sure
s 3.
5?10
tim
es
1 � 2
1 � 41 � 3
3 � 22 � 3
25 � 75 � 77 � 5
Pra
ctic
e (
Ave
rag
e)
Pro
per
ties
of
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 10-3)
Readin
g t
o L
earn
Math
em
ati
csP
rop
erti
es o
f L
og
arit
hm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
9G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Pre-
Act
ivit
yH
ow a
re t
he
pro
per
ties
of
exp
onen
ts a
nd
log
arit
hm
s re
late
d?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-3 a
t th
e to
p of
pag
e 54
1 in
you
r te
xtbo
ok.
Fin
d th
e va
lue
of l
og5
125.
3F
ind
the
valu
e of
log
55.
1F
ind
the
valu
e of
log
5(1
25
5).
2W
hic
h o
f th
e fo
llow
ing
stat
emen
ts i
s tr
ue?
BA
.lo
g 5(1
25
5) �
(log
512
5)
(log
55)
B.l
og5
(125
5)
�lo
g 512
5 �
log 5
5
Rea
din
g t
he
Less
on
1.E
ach
of
the
prop
erti
es o
f lo
gari
thm
s ca
n b
e st
ated
in
wor
ds o
r in
sym
bols
.Com
plet
e th
est
atem
ents
of
thes
e pr
oper
ties
in
wor
ds.
a.T
he
loga
rith
m o
f a
quot
ien
t is
th
e of
th
e lo
gari
thm
s of
th
e
and
the
.
b.
Th
e lo
gari
thm
of
a po
wer
is
the
of t
he
loga
rith
m o
f th
e ba
se a
nd
the
.
c.T
he
loga
rith
m o
f a
prod
uct
is
the
of t
he
loga
rith
ms
of i
ts
.
2.S
tate
wh
eth
er e
ach
of
the
foll
owin
g eq
uat
ion
s is
tru
eor
fal
se.I
f th
e st
atem
ent
is t
rue,
nam
e th
e pr
oper
ty o
f lo
gari
thm
s th
at i
s il
lust
rate
d.
a.lo
g 310
�lo
g 330
�lo
g 33
tru
e;Q
uo
tien
t P
rop
erty
b.l
og4
12 �
log 4
4
log 4
8fa
lse
c.lo
g 281
�2
log 2
9tr
ue;
Po
wer
Pro
per
tyd
.log
830
�lo
g 85
�lo
g 86
fals
e
3.T
he
alge
brai
c pr
oces
s of
sol
vin
g th
e eq
uat
ion
log
2x
lo
g 2(x
2)
�3
lead
s to
“x
��
4or
x�
2.”
Doe
s th
is m
ean
th
at b
oth
�4
and
2 ar
e so
luti
ons
of t
he
loga
rith
mic
equ
atio
n?
Exp
lain
you
r re
ason
ing.
Sam
ple
an
swer
:N
o;
2 is
a s
olu
tio
n b
ecau
se it
ch
ecks
:lo
g2
2 �
log
2(2
�2)
�lo
g2
2 �
log
24
�1
�2
�3.
Ho
wev
er,
bec
ause
log
2(�
4) a
nd
log
2(�
2) a
re u
nd
efin
ed,�
4 is
an
ext
ran
eou
sso
luti
on
an
d m
ust
be
elim
inat
ed.T
he
on
ly s
olu
tio
n is
2.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hing
is
to r
elat
e it
som
ethi
ng y
ou a
lrea
dy k
now
.Use
wor
dsto
exp
lain
how
th
e P
rodu
ct P
rope
rty
for
expo
nen
ts c
an h
elp
you
rem
embe
r th
e pr
odu
ctpr
oper
ty f
or l
ogar
ith
ms.
Sam
ple
an
swer
:Wh
en y
ou
mu
ltip
lytw
o n
um
ber
s o
rex
pre
ssio
ns
wit
h t
he
sam
e b
ase,
you
ad
dth
e ex
po
nen
ts a
nd
kee
p t
he
sam
e b
ase.
Lo
gar
ith
ms
are
exp
on
ents
,so
to
fin
d t
he
log
arit
hm
of
ap
rod
uct
,yo
u a
dd
the
log
arit
hm
s o
f th
e fa
cto
rs,k
eep
ing
th
e sa
me
bas
e.
fact
ors
sum
exp
on
ent
pro
du
ctd
eno
min
ato
rn
um
erat
or
dif
fere
nce
©G
lenc
oe/M
cGra
w-H
ill59
0G
lenc
oe A
lgeb
ra 2
Sp
iral
sC
onsi
der
an a
ngl
e in
sta
nda
rd p
osit
ion
wit
h i
ts v
erte
x at
a p
oin
t O
call
ed t
he
pole
.Its
in
itia
l si
de i
s on
a c
oord
inat
ized
axi
s ca
lled
th
e po
lar
axis
.A p
oin
t P
on t
he
term
inal
sid
e of
th
e an
gle
is n
amed
by
the
pola
r co
ord
inat
es(r
,�),
wh
ere
ris
th
e di
rect
ed d
ista
nce
of
the
poin
t fr
om O
and
�is
th
e m
easu
re o
fth
e an
gle.
Gra
phs
in t
his
sys
tem
may
be
draw
n o
n p
olar
coo
rdin
ate
pape
rsu
ch a
s th
e ki
nd
show
n b
elow
.
1.U
se a
cal
cula
tor
to c
ompl
ete
the
tabl
e fo
r lo
g 2r
�� 12�
0�.
(Hin
t:T
o fi
nd
�on
a c
alcu
lato
r,pr
ess
120
r2
.)
2.P
lot
the
poin
ts f
oun
d in
Exe
rcis
e 1
on t
he
grid
abo
ve a
nd
con
nec
t to
fo
rm a
sm
ooth
cu
rve.
Th
is t
ype
of s
pira
l is
cal
led
a lo
gari
thm
ic s
pira
l be
cau
se t
he
angl
e m
easu
res
are
prop
orti
onal
to
the
loga
rith
ms
of t
he
radi
i.
r1
23
45
67
8
�0�
120�
190�
240�
279�
310�
337�
360�
)
LOG
�)
LO
G�
01020
30
40
5060
7080
9010
011
012
013
0
140
150
160
170
180
190 200 21
0 220 23
0
240
250
260
270
280
290
300
310
32033
0340350
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
A
Answers (Lesson 10-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Co
mm
on
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
1G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Co
mm
on
Lo
gar
ith
ms
Bas
e 10
log
arit
hm
s ar
e ca
lled
com
mon
log
arit
hm
s.T
he
expr
essi
on l
og10
xis
usu
ally
wri
tten
wit
hou
t th
e su
bscr
ipt
as l
og x
.Use
th
e ke
y on
you
r ca
lcu
lato
r to
eva
luat
e co
mm
on l
ogar
ith
ms.
Th
e re
lati
on b
etw
een
exp
onen
ts a
nd
loga
rith
ms
give
s th
e fo
llow
ing
iden
tity
.
Inve
rse
Pro
per
ty o
f L
og
arit
hm
s an
d E
xpo
nen
ts10
log
x�
x
Eva
luat
e lo
g 50
to
fou
r d
ecim
al p
lace
s.U
se t
he
LO
G k
ey o
n y
our
calc
ula
tor.
To
fou
r de
cim
al p
lace
s,lo
g 50
�1.
6990
.
Sol
ve 3
2x�
1�
12.
32x
1
�12
Orig
inal
equ
atio
n
log
32x
1
�lo
g 12
Pro
pert
y of
Equ
ality
for
Log
arith
ms
(2x
1)
log
3 �
log
12P
ower
Pro
pert
y of
Log
arith
ms
2x
1 �
Div
ide
each
sid
e by
log
3.
2x�
�1
Sub
trac
t 1
from
eac
h si
de.
x�
��
1 �M
ultip
ly e
ach
side
by
.
x
0.63
09
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 18
2.lo
g 39
3.lo
g 12
01.
2553
1.59
112.
0792
4.lo
g 5.
85.
log
42.3
6.lo
g 0.
003
0.76
341.
6263
�2.
5229
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Rou
nd
to
fou
r d
ecim
al p
lace
s.
7.43
x�
120.
5975
8.6x
2
�18
�0.
3869
9.54
x�
2�
120
1.24
3710
.73x
�1
21
{x|x
0.
8549
}
11.2
.4x
4
�30
�0.
1150
12.6
.52x
20
0{x
|x
1.41
53}
13.3
.64x
�1
�85
.41.
1180
14.2
x
5�
3x�
213
.966
6
15.9
3x�
45x
2
�8.
1595
16.6
x�
5�
27x
3
�3.
6069
1 � 2lo
g 12
� log
31 � 2lo
g 12
� log
3
log
12� lo
g 3
LOG
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill59
2G
lenc
oe A
lgeb
ra 2
Ch
ang
e o
f B
ase
Form
ula
Th
e fo
llow
ing
form
ula
is
use
d to
ch
ange
exp
ress
ion
s w
ith
diff
eren
t lo
gari
thm
ic b
ases
to
com
mon
log
arit
hm
exp
ress
ion
s.
Ch
ang
e o
f B
ase
Fo
rmu
laF
or a
ll po
sitiv
e nu
mbe
rs a
, b,
and
n,
whe
re a
�1
and
b�
1, lo
g an
�
Exp
ress
log
815
in
ter
ms
of c
omm
on l
ogar
ith
ms.
Th
en a
pp
roxi
mat
eit
s va
lue
to f
our
dec
imal
pla
ces.
log 8
15�
Cha
nge
of B
ase
For
mul
a
1.
3023
Sim
plify
.
Th
e va
lue
of l
og8
15 i
s ap
prox
imat
ely
1.30
23.
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
1.lo
g 316
2.lo
g 240
3.lo
g 535
,2.5
237
,5.3
219
,2.2
091
4.lo
g 422
5.lo
g 12
200
6.lo
g 250
,2.2
297
,2.1
322
,5.6
439
7.lo
g 50.
48.
log 3
29.
log 4
28.5
,�0.
5693
,0.6
309
,2.4
164
10.l
og3
(20)
211
.log
6(5
)412
.log
8(4
)5
,5.4
537
,3.5
930
,3.3
333
13.l
og5
(8)3
14.l
og2
(3.6
)615
.log
12(1
0.5)
4
,3.8
761
,11.
0880
,3.7
851
16.l
og3
�15
0�
17.l
og4
3 �39�
18.l
og5
4 �16
00�
,2.2
804
,0.8
809
,1.1
460
log
160
0�
�4
log
5lo
g 3
9� 3
log
4lo
g 1
50� 2
log
3
4 lo
g 1
0.5
��
log
12
6 lo
g 3
.6�
�lo
g 2
3 lo
g 8
�lo
g 5
5 lo
g 4
�lo
g 8
4 lo
g 5
�lo
g 6
2 lo
g 2
0�
�lo
g 3
log
28.
5�
�lo
g 4
log
2� lo
g 3
log
0.4
�lo
g 5
log
50
� log
2lo
g 2
00� lo
g 1
2lo
g 2
2� lo
g 4
log
35
� log
5lo
g 4
0� lo
g 2
log
16
� log
3
log 10
15� lo
g 108
log b
n� lo
g ba
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Co
mm
on
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 10-4)
Skil
ls P
ract
ice
Co
mm
on
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
3G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 6
0.77
822.
log
151.
1761
3.lo
g 1.
10.
0414
4.lo
g 0.
3�
0.52
29
Use
th
e fo
rmu
la p
H �
�lo
g[H
�]
to f
ind
th
e p
H o
f ea
ch s
ub
stan
ce g
iven
its
con
cen
trat
ion
of
hyd
roge
n i
ons.
5.ga
stri
c ju
ices
:[H
]
�1.
0 �
10�
1m
ole
per
lite
r1.
0
6.to
mat
o ju
ice:
[H
] �
7.94
�10
�5
mol
e pe
r li
ter
4.1
7.bl
ood:
[H
] �
3.98
�10
�8
mol
e pe
r li
ter
7.4
8.to
oth
past
e:[H
]
�1.
26 �
10�
10m
ole
per
lite
r9.
9
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Rou
nd
to
fou
r d
ecim
al p
lace
s.
9.3x
�24
3{x
|x�
5}10
.16v
��v �
v
�
11.8
p�
501.
8813
12.7
y�
151.
3917
13.5
3b�
106
0.96
5914
.45k
�37
0.52
09
15.1
27p
�12
00.
2752
16.9
2m�
270.
75
17.3
r�
5�
4.1
6.28
4318
.8y
4
�15
{y|y
��
2.69
77}
19.7
.6d
3
�57
.2�
1.00
4820
.0.5
t�
8�
16.3
3.97
32
21.4
2x2
�84
�1.
0888
22.5
x2
1 �10
�0.
6563
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
23.l
og3
7;
1.77
1224
.log
566
;2.
6032
25.l
og2
35;
5.12
9326
.log
610
;1.
2851
log
1010
��
log
106
log
1035
��
log
102
log
1066
��
log
105
log
107
� log
103
1 � 21 � 4
©G
lenc
oe/M
cGra
w-H
ill59
4G
lenc
oe A
lgeb
ra 2
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 10
12.
0043
2.lo
g 2.
20.
3424
3.lo
g 0.
05�
1.30
10
Use
th
e fo
rmu
la p
H �
�lo
g[H
�]
to f
ind
th
e p
H o
f ea
ch s
ub
stan
ce g
iven
its
con
cen
trat
ion
of
hyd
roge
n i
ons.
4.m
ilk:
[H
] �
2.51
�10
�7
mol
e pe
r li
ter
6.6
5.ac
id r
ain
:[H
]
�2.
51 �
10�
6m
ole
per
lite
r5.
6
6.bl
ack
coff
ee:[
H
] �
1.0
�10
�5
mol
e pe
r li
ter
5.0
7.m
ilk
of m
agn
esia
:[H
]
�3.
16 �
10�
11m
ole
per
lite
r10
.5
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Rou
nd
to
fou
r d
ecim
al p
lace
s.
8.2x
�25
{x|x
�4.
6439
}9.
5a�
120
2.97
4610
.6z
�45
.62.
1319
11.9
m
100
{m|m
2.
0959
}12
.3.5
x�
47.9
3.08
8513
.8.2
y�
64.5
1.98
02
14.2
b
1�
7.31
{b|b
1.
8699
}15.
42x
�27
1.18
8716
.2a
�4
�82
.110
.359
3
17.9
z�
2�
38{z
|z�
3.65
55}
18.5
w
3�
17�
1.23
9619
.30x
2�
50�
1.07
25
20.5
x2�
3�
72�
2.37
8521
.42x
�9x
1
3.81
8822
.2n
1
�52
n�
10.
9117
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
23.l
og5
12;
1.54
4024
.log
832
;1.
6667
25.l
og11
9 ;
0.91
63
26.l
og2
18
;4.
1699
27.l
og9
6;
0.81
5528
.log
7�
8�;
29.H
OR
TIC
ULT
UR
ES
iber
ian
iri
ses
flou
rish
wh
en t
he
con
cen
trat
ion
of
hyd
roge
n i
ons
[H
]in
th
e so
il i
s n
ot l
ess
than
1.5
8 �
10�
8m
ole
per
lite
r.W
hat
is
the
pH o
f th
e so
il i
n w
hic
hth
ese
iris
es w
ill
flou
rish
?7.
8 o
r le
ss
30.A
CID
ITY
Th
e pH
of
vin
egar
is
2.9
and
the
pH o
f m
ilk
is 6
.6.H
ow m
any
tim
es g
reat
er i
sth
e h
ydro
gen
ion
con
cen
trat
ion
of
vin
egar
th
an o
f m
ilk?
abo
ut
5000
31.B
IOLO
GY
Th
ere
are
init
iall
y 10
00 b
acte
ria
in a
cu
ltu
re.T
he
nu
mbe
r of
bac
teri
a do
ubl
esea
ch h
our.
Th
e n
um
ber
of b
acte
ria
Npr
esen
t af
ter
th
ours
is
N�
1000
(2)t
.How
lon
g w
ill
it t
ake
the
cult
ure
to
incr
ease
to
50,0
00 b
acte
ria?
abo
ut
5.6
h
32.S
OU
ND
An
equ
atio
n f
or l
oudn
ess
Lin
dec
ibel
s is
giv
en b
y L
�10
log
R,w
her
e R
is t
he
sou
nd’
s re
lati
ve i
nte
nsi
ty.A
n a
ir-r
aid
sire
n c
an r
each
150
dec
ibel
s an
d je
t en
gin
e n
oise
can
rea
ch 1
20 d
ecib
els.
How
man
y ti
mes
gre
ater
is
the
rela
tive
in
ten
sity
of
the
air-
raid
sire
n t
han
th
at o
f th
e je
t en
gin
e n
oise
?10
00
log
108
� 2 lo
g10
7lo
g10
6�
�lo
g10
9lo
g10
18�
�lo
g10
2
log
109
��
log
1011
log
1032
��
log
108
log
1012
��
log
105
Pra
ctic
e (
Ave
rag
e)
Co
mm
on
Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
0.53
43
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
A
Answers (Lesson 10-4)
Readin
g t
o L
earn
Math
em
ati
csC
om
mo
n L
og
arit
hm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
5G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Pre-
Act
ivit
yW
hy
is a
log
arit
hm
ic s
cale
use
d t
o m
easu
re a
cid
ity?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-4 a
t th
e to
p of
pag
e 54
7 in
you
r te
xtbo
ok.
Wh
ich
su
bsta
nce
is
mor
e ac
idic
,mil
k or
tom
atoe
s?
tom
ato
es
Rea
din
g t
he
Less
on
1.R
hon
da u
sed
the
foll
owin
g ke
ystr
okes
to
ente
r an
exp
ress
ion
on
her
gra
phin
g ca
lcu
lato
r:
17
Th
e ca
lcu
lato
r re
turn
ed t
he
resu
lt 1
.230
4489
21.
Wh
ich
of
the
foll
owin
g co
ncl
usi
ons
are
corr
ect?
a,c,
and
d
a.T
he
base
10
loga
rith
m o
f 17
is
abou
t 1.
2304
.
b.
Th
e ba
se 1
7 lo
gari
thm
of
10 i
s ab
out
1.23
04.
c.T
he
com
mon
log
arit
hm
of
17 i
s ab
out
1.23
0449
.
d.
101.
2304
4892
1is
ver
y cl
ose
to 1
7.
e.T
he
com
mon
log
arit
hm
of
17 i
s ex
actl
y 1.
2304
4892
1.
2.M
atch
eac
h e
xpre
ssio
n f
rom
th
e fi
rst
colu
mn
wit
h a
n e
xpre
ssio
n f
rom
th
e se
con
d co
lum
nth
at h
as t
he
sam
e va
lue.
a.lo
g 22
ivi.
log 4
1
b.
log
12 ii
iii
.log
28
c.lo
g 31
iii
i.lo
g 10
12
d.
log 5
viv
.log
55
e.lo
g 10
00ii
v.lo
g 0.
1
3.C
alcu
lato
rs d
o n
ot h
ave
keys
for
fin
din
g ba
se 8
log
arit
hm
s di
rect
ly.H
owev
er,y
ou c
an u
se
a ca
lcu
lato
r to
fin
d lo
g 820
if
you
app
ly t
he
form
ula
.
Wh
ich
of
the
foll
owin
g ex
pres
sion
s ar
e eq
ual
to
log 8
20?
B a
nd
C
A.l
og20
8B
.C
.D
.
Hel
pin
g Y
ou
Rem
emb
er
4.S
omet
imes
it
is e
asie
r to
rem
embe
r a
form
ula
if
you
can
sta
te i
t in
wor
ds.S
tate
th
ech
ange
of
base
for
mu
la i
n w
ords
.S
amp
le a
nsw
er:T
o c
han
ge
the
log
arit
hm
of
an
um
ber
fro
m o
ne
bas
e to
an
oth
er,d
ivid
e th
e lo
g o
f th
e o
rig
inal
nu
mb
erin
th
e o
ld b
ase
by t
he
log
of
the
new
bas
e in
th
e o
ld b
ase.
log
8� lo
g 20
log
20� lo
g 8
log 10
20� lo
g 108
chan
ge
of
bas
e
1 � 5
ENTE
R)
LO
G
©G
lenc
oe/M
cGra
w-H
ill59
6G
lenc
oe A
lgeb
ra 2
Th
e S
lide
Ru
leB
efor
e th
e in
ven
tion
of
elec
tron
ic c
alcu
lato
rs,c
ompu
tati
ons
wer
e of
ten
perf
orm
ed o
n a
sli
de r
ule
.A s
lide
ru
le i
s ba
sed
on t
he
idea
of
loga
rith
ms.
It h
astw
o m
ovab
le r
ods
labe
led
wit
h C
an
d D
sca
les.
Eac
h o
f th
e sc
ales
is
loga
rith
mic
.
To
mu
ltip
ly 2
�3
on a
sli
de r
ule
,mov
e th
e C
rod
to
the
righ
t as
sh
own
belo
w.Y
ou c
an f
ind
2 �
3 by
add
ing
log
2 to
log
3,a
nd
the
slid
e ru
le a
dds
the
len
gth
s fo
r yo
u.T
he
dist
ance
you
get
is
0.77
8,or
th
e lo
gari
thm
of
6.
Fol
low
th
e st
eps
to m
ake
a sl
ide
rule
.
1.U
se g
raph
pap
er t
hat
has
sm
all
squ
ares
,su
ch a
s 10
squ
ares
to
the
inch
.Usi
ng
the
scal
es s
how
n a
t th
e ri
ght,
plot
th
e cu
rve
y�
log
xfo
r x
�1,
1.5,
and
the
wh
ole
nu
mbe
rs f
rom
2 t
hro
ugh
10.
Mak
e an
obv
iou
s h
eavy
dot
for
eac
h p
oin
t pl
otte
d.
2.Yo
u w
ill
nee
d tw
o st
rips
of
card
boar
d.A
5-
by-7
in
dex
card
,cu
t in
hal
f th
e lo
ng
way
,w
ill
wor
k fi
ne.
Tu
rn t
he
grap
h y
ou m
ade
in
Exe
rcis
e 1
side
way
s an
d u
se i
t to
mar
ka
loga
rith
mic
sca
le o
n e
ach
of
the
two
stri
ps.T
he
figu
re s
how
s th
e m
ark
for
2 be
ing
draw
n.
3.E
xpla
in h
ow t
o u
se a
sli
de r
ule
to
divi
de 8
by
2.L
ine
up
th
e 2
on
th
e C
sca
le w
ith
th
e 8
on
th
e D
sca
le.T
he
qu
oti
ent
is t
he
nu
mb
er o
n t
he
D s
cale
bel
ow
th
e 1
on
th
e C
sca
le.
0
0.1
0.2
0.3
y
1 2
11.
52
y =
log
x
0.1
0.2
12
1 21
CD
2 4
3 6
45
67
89
83
57
9
log
6
log
3lo
g 2
12
34
56
78
9
12
34
56
78
9
C D
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
1–2.
See
st
ud
ents
’wo
rk.
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 10-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill59
7G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
Th
e ir
rati
onal
nu
mbe
r e
2.
7182
8… o
ften
occ
urs
as t
he
base
for
exp
onen
tial
an
d lo
gari
thm
ic f
un
ctio
ns
that
des
crib
e re
al-w
orld
ph
enom
ena.
Nat
ura
l Bas
e e
As
nin
crea
ses,
�1
�nap
proa
ches
e
2.71
828…
.
ln x
�lo
g ex
Th
e fu
nct
ion
s y
�ex
and
y�
ln x
are
inve
rse
fun
ctio
ns.
Inve
rse
Pro
per
ty o
f B
ase
ean
d N
atu
ral L
og
arit
hm
sel
n x
�x
ln e
x�
x
Nat
ura
l ba
se e
xpre
ssio
ns
can
be
eval
uat
ed u
sin
g th
e ex
and
ln k
eys
on y
our
calc
ula
tor.
Eva
luat
e ln
168
5.U
se a
cal
cula
tor.
ln 1
685
7.
4295
Wri
te a
log
arit
hm
ic e
qu
atio
n e
qu
ival
ent
to e
2x�
7.e2
x�
7 →
log e
7 �
2xor
2x
�ln
7
Eva
luat
e ln
e18
.U
se t
he
Inve
rse
Pro
pert
y of
Bas
e e
and
Nat
ura
l L
ogar
ith
ms.
ln e
18�
18
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.ln
732
2.ln
84,
350
3.ln
0.7
354.
ln 1
006.
5958
11.3
427
�0.
3079
4.60
52
5.ln
0.0
824
6.ln
2.3
887.
ln 1
28,2
458.
ln 0
.006
14�
2.49
620.
8705
11.7
617
�5.
0929
Wri
te a
n e
qu
ival
ent
exp
onen
tial
or
loga
rith
mic
eq
uat
ion
.
9.e1
5�
x10
.e3x
�45
11.l
n 2
0 �
x12
.ln
x�
8ln
x�
153x
�ln
45
ex
�20
x�
e8
13.e
�5x
�0.
214
.ln
(4x
) �
9.6
15.e
8.2
�10
x16
.ln
0.0
002
�x
�5x
�ln
0.2
4x�
e9.
6ln
10x
�8.
2e
x�
0.00
02
Eva
luat
e ea
ch e
xpre
ssio
n.
17.l
n e
318
.eln
42
19.e
ln 0
.520
.ln
e16
.2
342
0.5
16.2
1 � n
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill59
8G
lenc
oe A
lgeb
ra 2
Equ
atio
ns
and
Ineq
ual
itie
s w
ith
ean
d ln
All
pro
pert
ies
of l
ogar
ith
ms
from
earl
ier
less
ons
can
be
use
d to
sol
ve e
quat
ion
s an
d in
equ
alit
ies
wit
h n
atu
ral
loga
rith
ms.
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
a.3e
2x
2 �
103e
2x
2 �
10O
rigin
al e
quat
ion
3e2x
�8
Sub
trac
t 2
from
eac
h si
de.
e2x
�D
ivid
e ea
ch s
ide
by 3
.
ln e
2x�
ln
Pro
pert
y of
Equ
ality
for
Log
arith
ms
2x�
ln
Inve
rse
Pro
pert
y of
Exp
onen
ts a
nd L
ogar
ithm
s
x�
ln
Mul
tiply
eac
h si
de b
y �1 2� .
x
0.49
04U
se a
cal
cula
tor.
b.
ln (
4x�
1) �
2
ln (
4x�
1) �
2O
rigin
al in
equa
lity
eln
(4x
�1)
�e2
Writ
e ea
ch s
ide
usin
g ex
pone
nts
and
base
e.
0 �
4x�
1 �
e2In
vers
e P
rope
rty
of E
xpon
ents
and
Log
arith
ms
1 �
4x�
e2
1A
dditi
on P
rope
rty
of I
nequ
aliti
es
�x
�(e
2
1)M
ultip
licat
ion
Pro
pert
y of
Ine
qual
ities
0.25
�x
�2.
0973
Use
a c
alcu
lato
r.
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
1.e4
x�
120
2.ex
�25
3.ex
�2
4
�21
1.19
69{x
|x
3.21
89}
4.83
32
4.ln
6x
4
5.ln
(x
3)
�5
��
26.
e�8x
�50
x
9.09
9717
.085
5{x
|x
�0.
4890
}
7.e4
x�
1�
3 �
128.
ln (
5x
3) �
3.6
9.2e
3x
5 �
20.
9270
6.71
96n
o s
olu
tio
n
10.6
3e
x
1�
2111
.ln
(2x
�5)
�8
12.l
n 5
x
ln 3
x�
90.
6094
1492
.979
0{x
|x�
23.2
423}
1 � 41 � 4
8 � 31 � 2
8 � 38 � 3
8 � 3
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
A
Answers (Lesson 10-5)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill59
9G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.e3
20.0
855
2.e�
20.
1353
3.ln
20.
6931
4.ln
0.0
9�
2.40
79
Wri
te a
n e
qu
ival
ent
exp
onen
tial
or
loga
rith
mic
eq
uat
ion
.
5.ex
�3
x�
ln 3
6.e4
�8x
4 �
ln 8
x
7.ln
15
�x
ex
�15
8.ln
x
0.69
31x
e0
.693
1
Eva
luat
e ea
ch e
xpre
ssio
n.
9.el
n 3
310
.eln
2x
2x
11.l
n e
�2.
5�
2.5
12.l
n e
yy
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
13.e
x
5{x
|x
1.60
94}
14.e
x�
3.2
{x|x
�1.
1632
}
15.2
ex�
1 �
111.
7918
16.5
ex
3 �
181.
0986
17.e
3x�
301.
1337
18.e
�4x
�10
{x|x
��
0.57
56}
19.e
5x
4 �
34{x
|x�
0.68
02}
20.1
�2e
2x�
�19
1.15
13
21.l
n 3
x�
22.
4630
22.l
n 8
x�
32.
5107
23.l
n (
x�
2) �
29.
3891
24.l
n (
x
3) �
1�
0.28
17
25.l
n (
x
3) �
451
.598
226
.ln
x
ln 2
x�
21.
9221
Skil
ls P
ract
ice
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
©G
lenc
oe/M
cGra
w-H
ill60
0G
lenc
oe A
lgeb
ra 2
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.e1
.54.
4817
2.ln
82.
0794
3.ln
3.2
1.16
324.
e�0.
60.
5488
5.e4
.266
.686
36.
ln 1
07.
e�2.
50.
0821
8.ln
0.0
37�
3.29
68
Wri
te a
n e
qu
ival
ent
exp
onen
tial
or
loga
rith
mic
eq
uat
ion
.
9.ln
50
�x
10.l
n 3
6 �
2x11
.ln
6
1.79
1812
.ln
9.3
2.
2300
ex
�50
e2x
�36
e1.7
918
6
e2.
2300
9.
3
13.e
x�
814
.e5
�10
x15
.e�
x�
416
.e2
�x
1
x�
ln 8
5 �
ln 1
0xx
��
ln 4
2 �
ln (
x�
1)
Eva
luat
e ea
ch e
xpre
ssio
n.
17.e
ln 1
212
18.e
ln 3
x3x
19.l
n e
�1
�1
20.l
n e
�2y
�2y
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
21.e
x�
922
.e�
x�
3123
.ex
�1.
124
.ex
�5.
8
{x|x
�2.
1972
}�
3.43
400.
0953
1.75
79
25.2
ex�
3 �
126
.5ex
1
7
27.4
ex
�19
28.�
3ex
10
�8
0.69
31{x
|x
0.18
23}
2.70
81{x
|x�
�0.
4055
}
29.e
3x�
830
.e�
4x�
531
.e0.
5x�
632
.2e5
x�
24
0.69
31�
0.40
243.
5835
0.49
70
33.e
2x
1 �
5534
.e3x
�5
�32
35.9
e2
x�
1036
.e�
3x
7
15
1.99
451.
2036
0{x
|x
�0.
6931
}
37.l
n 4
x�
338
.ln
(�
2x)
�7
39.l
n 2
.5x
�10
40.l
n (
x�
6) �
1
5.02
14�
548.
3166
8810
.586
38.
7183
41.l
n (
x
2) �
342
.ln
(x
3)
�5
43.l
n 3
x
ln 2
x�
944
.ln
5x
ln
x�
7
18.0
855
145.
4132
36.7
493
14.8
097
INV
ESTI
NG
For
Exe
rcis
es 4
5 an
d 4
6,u
se t
he
form
ula
for
con
tin
uou
sly
com
pou
nd
ed i
nte
rest
,A�
Per
t ,w
her
e P
is t
he
pri
nci
pal
,ris
th
e an
nu
al i
nte
rest
rate
,an
d t
is t
he
tim
e in
yea
rs.
45.I
f S
arit
a de
posi
ts $
1000
in
an
acc
oun
t pa
yin
g 3.
4% a
nn
ual
in
tere
st c
ompo
un
ded
con
tin
uou
sly,
wh
at i
s th
e ba
lan
ce i
n t
he
acco
un
t af
ter
5 ye
ars?
$118
5.30
46.H
ow l
ong
wil
l it
tak
e th
e ba
lan
ce i
n S
arit
a’s
acco
un
t to
rea
ch $
2000
?ab
ou
t 20
.4 y
r
47.R
AD
IOA
CTI
VE
DEC
AY
Th
e am
oun
t of
a r
adio
acti
ve s
ubs
tan
ce y
that
rem
ain
s af
ter
tye
ars
is g
iven
by
the
equ
atio
n y
�ae
kt,w
her
e a
is t
he
init
ial
amou
nt
pres
ent
and
kis
the
deca
y co
nst
ant
for
the
radi
oact
ive
subs
tan
ce.I
f a
�10
0,y
�50
,an
d k
��
0.03
5,fi
nd
t.ab
ou
t 19
.8 y
r
Pra
ctic
e (
Ave
rag
e)
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 10-5)
Readin
g t
o L
earn
Math
em
ati
csB
ase
ean
d N
atu
ral L
og
arit
hm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill60
1G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Pre-
Act
ivit
yH
ow i
s th
e n
atu
ral
bas
e e
use
d i
n b
ank
ing?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-5 a
t th
e to
p of
pag
e 55
4 in
you
r te
xtbo
ok.
Su
ppos
e th
at y
ou d
epos
it $
675
in a
sav
ings
acc
oun
t th
at p
ays
an a
nn
ual
inte
rest
rat
e of
5%
.In
eac
h c
ase
list
ed b
elow
,in
dica
te w
hic
h m
eth
od o
fco
mpo
un
din
g w
ould
res
ult
in
mor
e m
oney
in
you
r ac
cou
nt
at t
he
end
of o
ne
year
.a.
ann
ual
com
pou
ndi
ng
or m
onth
ly c
ompo
un
din
gm
on
thly
b.q
uar
terl
y co
mpo
un
din
g or
dai
ly c
ompo
un
din
gd
aily
c.da
ily
com
pou
ndi
ng
or c
onti
nu
ous
com
pou
ndi
ng
con
tin
uo
us
Rea
din
g t
he
Less
on
1.Ja
gdis
h e
nte
red
the
foll
owin
g ke
ystr
okes
in
his
cal
cula
tor:
5
Th
e ca
lcu
lato
r re
turn
ed t
he
resu
lt 1
.609
4379
12.W
hic
h o
f th
e fo
llow
ing
con
clu
sion
s ar
eco
rrec
t?d
an
d f
a.T
he
com
mon
log
arit
hm
of
5 is
abo
ut
1.60
94.
b.
Th
e n
atu
ral
loga
rith
m o
f 5
is e
xact
ly 1
.609
4379
12.
c.T
he
base
5 l
ogar
ith
m o
f e
is a
bou
t 1.
6094
.
d.
Th
e n
atu
ral
loga
rith
m o
f 5
is a
bou
t 1.
6094
38.
e.10
1.60
9437
912
is v
ery
clos
e to
5.
f.e1
.609
4379
12is
ver
y cl
ose
to 5
.
2.M
atch
eac
h e
xpre
ssio
n f
rom
th
e fi
rst
colu
mn
wit
h i
ts v
alu
e in
th
e se
con
d co
lum
n.S
ome
choi
ces
may
be
use
d m
ore
than
on
ce o
r n
ot a
t al
l.
a.el
n 5
IVI.
1
b.
ln 1
VII
.10
c.el
n e
VI
III.
�1
d.
ln e
5IV
IV.5
e.ln
eI
V.0
f.ln
��I
IIV
I.e
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
ethi
ng i
s to
exp
lain
it
to s
omeo
ne e
lse.
Sup
pose
tha
t yo
u ar
est
udy
ing
wit
h a
cla
ssm
ate
wh
o is
pu
zzle
d w
hen
ask
ed t
o ev
alu
ate
ln e
3 .H
ow w
ould
you
expl
ain
to
him
an
eas
y w
ay t
o fi
gure
th
is o
ut?
Sam
ple
an
swer
:ln
mea
ns
nat
ura
llo
g.T
he
nat
ura
l lo
g o
f e
3is
th
e p
ow
er t
o w
hic
h y
ou
rai
se e
to g
et e
3 .T
his
is o
bvio
usl
y 3.
1 � e
ENTE
R)
LN
©G
lenc
oe/M
cGra
w-H
ill60
2G
lenc
oe A
lgeb
ra 2
Ap
pro
xim
atio
ns
for
�an
d e
Th
e fo
llow
ing
expr
essi
on c
an b
e u
sed
to a
ppro
xim
ate
e.If
gre
ater
an
d gr
eate
rva
lues
of
nar
e u
sed,
the
valu
e of
th
e ex
pres
sion
app
roxi
mat
es e
mor
e an
dm
ore
clos
ely.
�1
� n1 � �n
An
oth
er w
ay t
o ap
prox
imat
e e
is t
o u
se t
his
in
fin
ite
sum
.Th
e gr
eate
r th
eva
lue
of n
,th
e cl
oser
th
e ap
prox
imat
ion
.
e�
1
1
�1 2�
� 21 �
3�
� 2�
1 3�
4�
…
� 2
�3
�41
�…
�n
�
…
In a
sim
ilar
man
ner
,�ca
n b
e ap
prox
imat
ed u
sin
g an
in
fin
ite
prod
uct
disc
over
ed b
y th
e E
ngl
ish
mat
hem
atic
ian
Joh
n W
alli
s (1
616–
1703
).
�� 2��
�2 1��
�2 3��
�4 3��
�4 5��
�6 5��
�6 7��
… �
� 2n2 �n
1�
� � 2n
2 n1
�…
Sol
ve e
ach
pro
ble
m.
1.U
se a
cal
cula
tor
wit
h a
n e
xke
y to
fin
d e
to 7
dec
imal
pla
ces.
2.71
8281
8
2.U
se t
he
expr
essi
on �1
� n1 � �n
to a
ppro
xim
ate
eto
3 d
ecim
al p
lace
s.U
se
5,10
0,50
0,an
d 70
00 a
s va
lues
of
n.
2.48
8,2.
705,
2.71
6,2.
718
3.U
se t
he
infi
nit
e su
m t
o ap
prox
imat
e e
to 3
dec
imal
pla
ces.
Use
th
e w
hol
e n
um
bers
fro
m 3
th
rou
gh 6
as
valu
es o
f n
.2.
667,
2.70
8,2.
717,
2.71
8
4.W
hic
h a
ppro
xim
atio
n m
eth
od a
ppro
ach
es t
he
valu
e of
em
ore
quic
kly?
the
infi
nit
e su
m5.
Use
a c
alcu
lato
r w
ith
a �
key
to fi
nd
�to
7 d
ecim
al p
lace
s.3.
1415
927
6.U
se t
he
infi
nit
e pr
odu
ct t
o ap
prox
imat
e �
to 3
dec
imal
pla
ces.
Use
th
e w
hol
e n
um
bers
fro
m 3
th
rou
gh 6
as
valu
es o
f n
.2.
926,
2.97
2,3.
002,
3.02
3
7.D
oes
the
infi
nit
e pr
odu
ct g
ive
good
app
roxi
mat
ion
s fo
r �
quic
kly?
no
8.S
how
th
at �
4
�5
is e
qual
to
e6to
4 d
ecim
al p
lace
s.To
4 d
ecim
al p
lace
s,th
ey b
oth
eq
ual
403
.428
8.9.
Wh
ich
is
larg
er,e
�or
�e ?
e�>
�e
10.T
he
expr
essi
on x
reac
hes
a m
axim
um
val
ue
at x
�e.
Use
th
is f
act
to
prov
e th
e in
equ
alit
y yo
u f
oun
d in
Exe
rcis
e 9.
e�1 e�>
�� �1 � ;
� e�1 e� ��
e>
� �� �1 � ��
e ;e�
> �
e
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
A
Answers (Lesson 10-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Exp
on
enti
al G
row
th a
nd
Dec
ay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
3G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Exp
on
enti
al D
ecay
Dep
reci
atio
n o
f va
lue
and
radi
oact
ive
deca
y ar
e ex
ampl
es o
fex
pon
enti
al d
ecay
.Wh
en a
qu
anti
ty d
ecre
ases
by
a fi
xed
perc
ent
each
tim
e pe
riod
,th
eam
oun
t of
th
e qu
anti
ty a
fter
tti
me
peri
ods
is g
iven
by
y�
a(1
�r)
t ,w
her
e a
is t
he
init
ial
amou
nt
and
ris
th
e pe
rcen
t de
crea
se e
xpre
ssed
as
a de
cim
al.
An
oth
er e
xpon
enti
al d
ecay
mod
el o
ften
use
d by
sci
enti
sts
is y
�ae
�kt
,wh
ere
kis
a c
onst
ant.
CO
NSU
MER
PR
ICES
As
tech
nol
ogy
adva
nce
s,th
e p
rice
of
man
yte
chn
olog
ical
dev
ices
su
ch a
s sc
ien
tifi
c ca
lcu
lato
rs a
nd
cam
cord
ers
goes
dow
n.
On
e b
ran
d o
f h
and
-hel
d o
rgan
izer
sel
ls f
or $
89.
a.If
its
pri
ce d
ecre
ases
by
6% p
er y
ear,
how
mu
ch w
ill
it c
ost
afte
r 5
year
s?U
se t
he
expo
nen
tial
dec
ay m
odel
wit
h i
nit
ial
amou
nt
$89,
perc
ent
decr
ease
0.0
6,an
dti
me
5 ye
ars.
y�
a(1
�r)
tE
xpon
entia
l dec
ay f
orm
ula
y�
89(1
�0.
06)5
a�
89,
r�
0.06
, t
�5
y�
$65.
32A
fter
5 y
ears
th
e pr
ice
wil
l be
$65
.32.
b.
Aft
er h
ow m
any
year
s w
ill
its
pri
ce b
e $5
0?To
fin
d w
hen
the
pric
e w
ill b
e $5
0,ag
ain
use
the
expo
nent
ial d
ecay
for
mul
a an
d so
lve
for
t.y
�a(
1 �
r)t
Exp
onen
tial d
ecay
for
mul
a
50 �
89(1
�0.
06)t
y�
50,
a�
89,
r�
0.06
�(0
.94)
tD
ivid
e ea
ch s
ide
by 8
9.
log
���
log
(0.9
4)t
Pro
pert
y of
Equ
ality
for
Log
arith
ms
log
���
tlo
g 0.
94P
ower
Pro
pert
y
t�
Div
ide
each
sid
e by
log
0.94
.
t
9.3
Th
e pr
ice
wil
l be
$50
aft
er a
bou
t 9.
3 ye
ars.
1.B
USI
NES
SA
fu
rnit
ure
sto
re i
s cl
osin
g ou
t it
s bu
sin
ess.
Eac
h w
eek
the
own
er l
ower
spr
ices
by
25%
.Aft
er h
ow m
any
wee
ks w
ill
the
sale
pri
ce o
f a
$500
ite
m d
rop
belo
w $
100?
6 w
eeks
CA
RB
ON
DA
TIN
GU
se t
he
form
ula
y�
ae�
0.00
012t
,wh
ere
ais
th
e in
itia
l am
oun
t of
Car
bon
-14,
tis
th
e n
um
ber
of
year
s ag
o th
e an
imal
liv
ed,a
nd
yis
th
e re
mai
nin
gam
oun
t af
ter
tye
ars.
2.H
ow o
ld is
a f
ossi
l rem
ain
that
has
lost
95%
of
its
Car
bon-
14?
abo
ut
25,0
00 y
ears
old
3.H
ow o
ld is
a s
kele
ton
that
has
95%
of
its
Car
bon-
14 r
emai
ning
?ab
ou
t 42
7.5
year
s o
ld
log
��5 80 9��
��
log
0.94
50 � 8950 � 8950 � 89
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill60
4G
lenc
oe A
lgeb
ra 2
Exp
on
enti
al G
row
thP
opu
lati
on i
ncr
ease
an
d gr
owth
of
bact
eria
col
onie
s ar
e ex
ampl
esof
exp
onen
tial
gro
wth
.Wh
en a
qu
anti
ty i
ncr
ease
s by
a f
ixed
per
cen
t ea
ch t
ime
peri
od,t
he
amou
nt
of t
hat
qu
anti
ty a
fter
tti
me
peri
ods
is g
iven
by
y�
a(1
r)
t ,w
her
e a
is t
he
init
ial
amou
nt
and
ris
th
e pe
rcen
t in
crea
se (
or r
ate
of g
row
th)
expr
esse
d as
a d
ecim
al.
An
oth
er e
xpon
enti
al g
row
th m
odel
oft
en u
sed
by s
cien
tist
s is
y�
aekt
,wh
ere
kis
a c
onst
ant.
A c
omp
ute
r en
gin
eer
is h
ired
for
a s
alar
y of
$28
,000
.If
she
gets
a5%
rai
se e
ach
yea
r,af
ter
how
man
y ye
ars
wil
l sh
e b
e m
akin
g $5
0,00
0 or
mor
e?U
se t
he
expo
nen
tial
gro
wth
mod
el w
ith
a�
28,0
00,y
�50
,000
,an
d r
�0.
05 a
nd
solv
e fo
r t.
y�
a(1
r)
tE
xpon
entia
l gro
wth
for
mul
a
50,0
00 �
28,0
00(1
0.
05)t
y�
50,0
00,
a�
28,0
00,
r�
0.05
�(1
.05)
tD
ivid
e ea
ch s
ide
by 2
8,00
0.
log
���
log
(1.0
5)t
Pro
pert
y of
Equ
ality
of
Loga
rithm
s
log
���
tlo
g 1.
05P
ower
Pro
pert
y
t�
Div
ide
each
sid
e by
log
1.05
.
t
11.9
yea
rsU
se a
cal
cula
tor.
If r
aise
s ar
e gi
ven
an
nu
ally
,sh
e w
ill
be m
akin
g ov
er $
50,0
00 i
n 1
2 ye
ars.
1.B
AC
TER
IA G
RO
WTH
A c
erta
in s
trai
n o
f ba
cter
ia g
row
s fr
om 4
0 to
326
in
120
min
ute
s.F
ind
kfo
r th
e gr
owth
for
mu
la y
�ae
kt,w
her
e t
is i
n m
inu
tes.
abo
ut
0.01
75
2.IN
VES
TMEN
TC
arl
plan
s to
in
vest
$50
0 at
8.2
5% i
nte
rest
,com
pou
nde
d co
nti
nu
ousl
y.H
ow l
ong
wil
l it
tak
e fo
r h
is m
oney
to
trip
le?
abo
ut
14 y
ears
3.SC
HO
OL
POPU
LATI
ON
Th
ere
are
curr
entl
y 85
0 st
ude
nts
at
the
hig
h s
choo
l,w
hic
hre
pres
ents
fu
ll c
apac
ity.
Th
e to
wn
pla
ns
an a
ddit
ion
to
hou
se 4
00 m
ore
stu
den
ts.I
f th
e sc
hoo
l po
pula
tion
gro
ws
at 7
.8%
per
yea
r,in
how
man
y ye
ars
wil
l th
e n
ew a
ddit
ion
be f
ull
?ab
ou
t 5
year
s
4.EX
ERC
ISE
Hu
go b
egin
s a
wal
kin
g pr
ogra
m b
y w
alki
ng
mil
e pe
r da
y fo
r on
e w
eek.
Eac
h w
eek
ther
eaft
er h
e in
crea
ses
his
mil
eage
by
10%
.Aft
er h
ow m
any
wee
ks i
s h
ew
alki
ng
mor
e th
an 5
mil
es p
er d
ay?
24 w
eeks
5.V
OC
AB
ULA
RY G
RO
WTH
Wh
en E
mil
y w
as 1
8 m
onth
s ol
d,sh
e h
ad a
10-
wor
dvo
cabu
lary
.By
the
tim
e sh
e w
as 5
yea
rs o
ld (
60 m
onth
s),h
er v
ocab
ular
y w
as 2
500
wor
ds.
If h
er v
ocab
ular
y in
crea
sed
at a
con
stan
t pe
rcen
t pe
r m
onth
,wha
t w
as t
hat
incr
ease
?ab
ou
t 14
%
1 � 2
log
��5 20 8��
� log
1.05
50 � 2850 � 2850 � 28
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Exp
on
enti
al G
row
th a
nd
Dec
ay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 10-6)
Skil
ls P
ract
ice
Exp
on
enti
al G
row
th a
nd
Dec
ay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
5G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Sol
ve e
ach
pro
ble
m.
1.FI
SHIN
GIn
an
ove
r-fi
shed
are
a,th
e ca
tch
of
a ce
rtai
n f
ish
is
decr
easi
ng
at a
n a
vera
gera
te o
f 8%
per
yea
r.If
th
is d
ecli
ne
pers
ists
,how
lon
g w
ill
it t
ake
for
the
catc
h t
o re
ach
hal
f of
th
e am
oun
t be
fore
th
e de
clin
e?ab
ou
t 8.
3 yr
2.IN
VES
TIN
GA
lex
inve
sts
$200
0 in
an
acc
oun
t th
at h
as a
6%
an
nu
al r
ate
of g
row
th.T
oth
e n
eare
st y
ear,
wh
en w
ill
the
inve
stm
ent
be w
orth
$36
00?
10 y
r
3.PO
PULA
TIO
NA
cu
rren
t ce
nsu
s sh
ows
that
th
e po
pula
tion
of
a ci
ty i
s 3.
5 m
illi
on.U
sin
gth
e fo
rmu
la P
�ae
rt,f
ind
the
expe
cted
pop
ula
tion
of
the
city
in
30
year
s if
th
e gr
owth
rate
rof
th
e po
pula
tion
is
1.5%
per
yea
r,a
repr
esen
ts t
he
curr
ent
popu
lati
on i
n m
illi
ons,
and
tre
pres
ents
th
e ti
me
in y
ears
.ab
ou
t 5.
5 m
illio
n
4.PO
PULA
TIO
NT
he
popu
lati
on P
in t
hou
san
ds o
f a
city
can
be
mod
eled
by
the
equ
atio
nP
�80
e0.0
15t ,
wh
ere
tis
th
e ti
me
in y
ears
.In
how
man
y ye
ars
wil
l th
e po
pula
tion
of
the
city
be
120,
000?
abo
ut
27 y
r
5.B
AC
TER
IAH
ow m
any
days
wil
l it
tak
e a
cult
ure
of
bact
eria
to
incr
ease
fro
m 2
000
to50
,000
if
the
grow
th r
ate
per
day
is 9
3.2%
?ab
ou
t 4.
9 d
ays
6.N
UC
LEA
R P
OW
ERT
he
elem
ent
plu
ton
ium
-239
is
hig
hly
rad
ioac
tive
.Nu
clea
r re
acto
rsca
n p
rodu
ce a
nd
also
use
th
is e
lem
ent.
Th
e h
eat
that
plu
ton
ium
-239
em
its
has
hel
ped
topo
wer
equ
ipm
ent
on t
he
moo
n.I
f th
e h
alf-
life
of
plu
ton
ium
-239
is
24,3
60 y
ears
,wh
at i
sth
e va
lue
of k
for
this
ele
men
t?ab
ou
t 0.
0000
2845
7.D
EPR
ECIA
TIO
NA
Glo
bal
Pos
itio
nin
g S
atel
lite
(G
PS
) sy
stem
use
s sa
tell
ite
info
rmat
ion
to l
ocat
e gr
oun
d po
siti
on.A
bu’s
su
rvey
ing
firm
bou
ght
a G
PS
sys
tem
for
$12
,500
.Th
eG
PS
dep
reci
ated
by
a fi
xed
rate
of
6% a
nd
is n
ow w
orth
$86
00.H
ow l
ong
ago
did
Abu
buy
the
GP
S s
yste
m?
abo
ut
6.0
yr
8.B
IOLO
GY
In a
lab
orat
ory,
an o
rgan
ism
gro
ws
from
100
to
250
in 8
hou
rs.W
hat
is
the
hou
rly
grow
th r
ate
in t
he
grow
th f
orm
ula
y�
a(1
r)
t ?ab
ou
t 12
.13%
©G
lenc
oe/M
cGra
w-H
ill60
6G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
pro
ble
m.
1.IN
VES
TIN
GT
he f
orm
ula
A�
P�1
�2t
give
s th
e va
lue
of a
n in
vest
men
t af
ter
tye
ars
in
an a
ccou
nt
that
ear
ns
an a
nn
ual
in
tere
st r
ate
rco
mpo
un
ded
twic
e a
year
.Su
ppos
e $5
00is
in
vest
ed a
t 6%
an
nu
al i
nte
rest
com
pou
nde
d tw
ice
a ye
ar.I
n h
ow m
any
year
s w
ill
the
inve
stm
ent
be w
orth
$10
00?
abo
ut
11.7
yr
2.B
AC
TER
IAH
ow m
any
hou
rs w
ill
it t
ake
a cu
ltu
re o
f ba
cter
ia t
o in
crea
se f
rom
20
to20
00 i
f th
e gr
owth
rat
e pe
r h
our
is 8
5%?
abo
ut
7.5
h
3.R
AD
IOA
CTI
VE
DEC
AY
A r
adio
acti
ve s
ubs
tan
ce h
as a
hal
f-li
fe o
f 32
yea
rs.F
ind
the
con
stan
t k
in t
he
deca
y fo
rmu
la f
or t
he
subs
tan
ce.
abo
ut
0.02
166
4.D
EPR
ECIA
TIO
NA
pie
ce o
f m
ach
iner
y va
lued
at
$250
,000
dep
reci
ates
at
a fi
xed
rate
of
12%
per
yea
r.A
fter
how
man
y ye
ars
wil
l th
e va
lue
hav
e de
prec
iate
d to
$10
0,00
0?ab
ou
t 7.
2 yr
5.IN
FLA
TIO
NFo
r D
ave
to b
uy a
new
car
com
para
bly
equi
pped
to
the
one
he b
ough
t 8
year
sag
o w
ould
cos
t $1
2,50
0.S
ince
Dav
e bo
ugh
t th
e ca
r,th
e in
flat
ion
rat
e fo
r ca
rs l
ike
his
has
been
at
an a
vera
ge a
nn
ual
rat
e of
5.1
%.I
f D
ave
orig
inal
ly p
aid
$840
0 fo
r th
e ca
r,h
owlo
ng
ago
did
he
buy
it?
abo
ut
8 yr
6.R
AD
IOA
CTI
VE
DEC
AY
Cob
alt,
an e
lem
ent
use
d to
mak
e al
loys
,has
sev
eral
iso
tope
s.O
ne
of t
hes
e,co
balt
-60,
is r
adio
acti
ve a
nd
has
a h
alf-
life
of
5.7
year
s.C
obal
t-60
is
use
d to
trac
e th
e pa
th o
f n
onra
dioa
ctiv
e su
bsta
nce
s in
a s
yste
m.W
hat
is
the
valu
e of
kfo
rC
obal
t-60
?ab
ou
t 0.
1216
7.W
HA
LES
Mod
ern
wh
ales
app
eare
d 5�
10 m
illi
on y
ears
ago
.Th
e ve
rteb
rae
of a
wh
ale
disc
over
ed b
y pa
leon
tolo
gist
s co
nta
in r
ough
ly 0
.25%
as
mu
ch c
arbo
n-1
4 as
th
ey w
ould
hav
e co
nta
ined
wh
en t
he
wh
ale
was
ali
ve.H
ow l
ong
ago
did
the
wh
ale
die?
Use
k
�0.
0001
2.ab
ou
t 50
,000
yr
8.PO
PULA
TIO
NT
he
popu
lati
on o
f ra
bbit
s in
an
are
a is
mod
eled
by
the
grow
th e
quat
ion
P(t
) �
8e0.
26t ,
wh
ere
Pis
in
th
ousa
nds
an
d t
is i
n y
ears
.How
lon
g w
ill
it t
ake
for
the
popu
lati
on t
o re
ach
25,
000?
abo
ut
4.4
yr
9.D
EPR
ECIA
TIO
NA
com
pute
r sy
stem
dep
reci
ates
at
an a
vera
ge r
ate
of 4
% p
er m
onth
.If
the
valu
e of
th
e co
mpu
ter
syst
em w
as o
rigi
nal
ly $
12,0
00,i
n h
ow m
any
mon
ths
is i
tw
orth
$73
50?
abo
ut
12 m
o
10.B
IOLO
GY
In a
lab
orat
ory,
a cu
ltu
re i
ncr
ease
s fr
om 3
0 to
195
org
anis
ms
in 5
hou
rs.
Wh
at i
s th
e h
ourl
y gr
owth
rat
e in
th
e gr
owth
for
mu
la y
�a
(1
r)t ?
abo
ut
45.4
%
r � 2
Pra
ctic
e (
Ave
rag
e)
Exp
on
enti
al G
row
th a
nd
Dec
ay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
A
Answers (Lesson 10-6)
Readin
g t
o L
earn
Math
em
ati
csE
xpo
nen
tial
Gro
wth
an
d D
ecay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
7G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e th
e cu
rren
t va
lue
of y
our
car?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-6 a
t th
e to
p of
pag
e 56
0 in
you
r te
xtbo
ok.
•B
etw
een
wh
ich
tw
o ye
ars
show
n i
n t
he
tabl
e di
d th
e ca
r de
prec
iate
by
the
grea
test
am
oun
t?b
etw
een
yea
rs 0
an
d 1
•D
escr
ibe
two
way
s to
cal
cula
te t
he
valu
e of
th
e ca
r 6
year
s af
ter
it w
aspu
rch
ased
.(D
o n
ot a
ctu
ally
cal
cula
te t
he
valu
e.)
Sam
ple
an
swer
:1.
Mu
ltip
ly $
9200
.66
by 0
.16
and
su
btr
act
the
resu
lt f
rom
$92
00.6
6.2.
Mu
ltip
ly $
9200
.66
by 0
.84.
Rea
din
g t
he
Less
on
1.S
tate
wh
eth
er e
ach
sit
uat
ion
is
an e
xam
ple
of e
xpon
enti
al g
row
thor
dec
ay.
a.A
cit
y h
ad 4
2,00
0 re
side
nts
in
198
0 an
d 12
8,00
0 re
side
nts
in
200
0.g
row
th
b.
Rau
l co
mpa
red
the
valu
e of
his
car
wh
en h
e bo
ugh
t it
new
to
the
valu
e w
hen
he
trad
ed ‘;
lpit
in
six
yea
rs l
ater
.d
ecay
c.A
pal
eon
tolo
gist
com
pare
d th
e am
oun
t of
car
bon
-14
in t
he
skel
eton
of
an a
nim
alw
hen
it
died
to
the
amou
nt
300
year
s la
ter.
dec
ay
d.
Mar
ia d
epos
ited
$75
0 in
a s
avin
gs a
ccou
nt
payi
ng
4.5%
an
nu
al i
nte
rest
com
pou
nde
dqu
arte
rly.
Sh
e di
d n
ot m
ake
any
wit
hdr
awal
s or
fu
rth
er d
epos
its.
Sh
e co
mpa
red
the
bala
nce
in
her
pas
sboo
k im
med
iate
ly a
fter
sh
e op
ened
th
e ac
cou
nt
to t
he
bala
nce
3
year
s la
ter.
gro
wth
2.S
tate
wh
eth
er e
ach
equ
atio
n r
epre
sen
ts e
xpon
enti
al g
row
th o
r de
cay.
a.y
�5e
0.15
tg
row
thb
.y�
1000
(1 �
0.05
)td
ecay
c.y
�0.
3e�
1200
td
ecay
d.y
�2(
1
0.00
01)t
gro
wth
Hel
pin
g Y
ou
Rem
emb
er
3.V
isu
aliz
ing
thei
r gr
aph
s is
oft
en a
goo
d w
ay t
o re
mem
ber
the
diff
eren
ce b
etw
een
mat
hem
atic
al e
quat
ions
.How
can
you
r kn
owle
dge
of t
he g
raph
s of
exp
onen
tial
equ
atio
nsfr
om L
esso
n 1
0-1
hel
p yo
u t
o re
mem
ber
that
equ
atio
ns
of t
he
form
y�
a(1
r)
t
repr
esen
t ex
pon
enti
al g
row
th,w
hil
e eq
uat
ion
s of
th
e fo
rm y
�a(
1 �
r)t
repr
esen
tex
pon
enti
al d
ecay
?S
amp
le a
nsw
er:
If a
�0,
the
gra
ph
of
y�
abx
is a
lway
s in
crea
sin
g if
b
�1
and
is a
lway
s d
ecre
asin
g if
0 �
b�
1.S
ince
ris
alw
ays
a p
osi
tive
nu
mb
er,i
f b
�1
�r,
the
bas
e w
ill b
e g
reat
er t
han
1 a
nd
th
e fu
nct
ion
will
be
incr
easi
ng
(g
row
th),
wh
ile if
b�
1 �
r,th
e b
ase
will
be
less
th
an 1
and
th
e fu
nct
ion
will
be
dec
reas
ing
(d
ecay
).
©G
lenc
oe/M
cGra
w-H
ill60
8G
lenc
oe A
lgeb
ra 2
Eff
ecti
ve A
nn
ual
Yie
ldW
hen
in
tere
st i
s co
mpo
un
ded
mor
e th
an o
nce
per
yea
r,th
e ef
fect
ive
ann
ual
yiel
d is
hig
her
th
an t
he
ann
ual
in
tere
st r
ate.
Th
e ef
fect
ive
ann
ual
yie
ld,E
,is
the
inte
rest
rat
e th
at w
ould
giv
e th
e sa
me
amou
nt
of i
nte
rest
if
the
inte
rest
wer
e co
mpo
un
ded
once
per
yea
r.If
Pdo
llar
s ar
e in
vest
ed f
or o
ne
year
,th
eva
lue
of t
he
inve
stm
ent
at t
he
end
of t
he
year
is
A�
P(1
E
).If
Pdo
llar
sar
e in
vest
ed f
or o
ne
year
at
a n
omin
al r
ate
rco
mpo
un
ded
nti
mes
per
yea
r,
the
valu
e of
th
e in
vest
men
t at
th
e en
d of
th
e ye
ar i
s A
�P�1
� nr � �n
.Set
tin
g
the
amou
nts
equ
al a
nd
solv
ing
for
Ew
ill
prod
uce
a f
orm
ula
for
th
e ef
fect
ive
ann
ual
yie
ld.
P(1
E
) �
P�1
� nr � �n
1
E�
�1
� nr � �n
E�
�1
� nr � �n�
1
If c
ompo
un
din
g is
con
tin
uou
s,th
e va
lue
of t
he
inve
stm
ent
at t
he
end
of o
ne
year
is
A�
Per
.Aga
in s
et t
he
amou
nts
equ
al a
nd
solv
e fo
r E
.A f
orm
ula
for
the
effe
ctiv
e an
nu
al y
ield
un
der
con
tin
uou
s co
mpo
un
din
g is
obt
ain
ed.
P(1
E
) �
Per
1
E�
er
E�
er�
1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
Fin
d t
he
effe
ctiv
ean
nu
al y
ield
of
an i
nve
stm
ent
mad
e at
7.5%
com
pou
nd
ed m
onth
ly.
r�
0.07
5
n�
12
E�
�1
�0.10 275 �
�12�
1 �
7.7
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Fin
d t
he
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ctiv
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al y
ield
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e at
6.25
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omp
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tin
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25
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625
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� 6
.45%
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Fin
d t
he
effe
ctiv
e an
nu
al y
ield
for
eac
h i
nve
stm
ent.
1.10
% c
ompo
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ded
quar
terl
y10
.38%
2.8.
5% c
ompo
un
ded
mon
thly
8.84
%
3.9.
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com
pou
nde
d co
nti
nu
ousl
y9.
69%
4.7.
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com
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d co
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y8.
06%
5.6.
5% c
ompo
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ded
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y (a
ssu
me
a 36
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y ye
ar)
6.72
%
6.W
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h i
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ent
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ds m
ore
inte
rest
—9%
com
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d co
nti
nu
ousl
y or
9.
2% c
ompo
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ded
quar
terl
y?9.
2% q
uar
terl
y
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