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Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

1. Graph exponential functions. (Sections 7.1, 7.2) Worksheet 1 1 – 36

2. Solve exponential growth and exponential decay problems. (Sections 7.1, 7.2) Worksheet 2 1 – 18

3. Simplify logarithmic expressions. (Section 7.4) Page 503 8 – 19 Worksheet 3 1 – 37

4. Common and natural logs, inverse properties of log wx x and

log x wx , log 1b ,

graph logarithmic equations. Worksheet 4 1 – 39

5. Apply the 3 laws (properties) of logs. (Section 7.5) Page 510 15 – 44 Worksheet 5 1 – 30

6. Approximating logarithmic values. Change of base theorem. Worksheet 6 1 – 23

7. Solve exponential equations with a common base. Worksheet 7 1-20

Page 519 3 – 11 8. Solve logarithmic equations (Section 7.6) Worksheet 8 1 – 34

9. Solve log equations. (Section 7.6)

Worksheet 9 1 – 33

10. Solve exponential equations without a common base.

Worksheet 10 1-13

Review Review Worksheet 1 1 – 90 Review Worksheet 2 1 – 53 Review Worksheet 3 1 - 13

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Algebra 2 Unit 8 Worksheet 1 CALCULATORS ARE NOT ALLOWED

Simplify:

1. 43 2. 32 − 3. 21

5

4.

121

4

−

5. 07 6. 1 4 7. 41

2

−

8. 29 −

9. 5 -2 10. 1236 11.

2364 12.

1216

−

Definition:

The function defined by y = b x is called an exponential function with base b

Requirements: b > 0, b ≠ 1

Characteristics of exponential functions:

The basic graph of an exponential function looks like the following: An increasing exponential if they rise as they go from left to right. A decreasing exponential if they drop as they go from left to right. Other characteristics: The x-axis is a horizontal asymptote of the graph and the graphs contain the point (0,1). In problems 13 – 16, complete the table of values and then graph on graph paper.

13. y = 2 x 14. y = 3 x 15. 12

=

x

y 16. 15

=

x

y

x y

2

1

0

–1

–2

Increasing

Decreasing

x y

2

1

0

–1

–2

x y

2

1

0

–1

–2

x y

2

1

0

–1

–2

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Sketch the following graphs of the exponential functions and state if they are

increasing or decreasing graphs. Be sure to label the intercepts.

17. y = 4 x 18. 13

x

y =

19. y = 5 x 20. 14

x

y =

Create a table of values in problems 21 -23 and then graph on graph paper. 21. y = 1 x 22. y = 0 x 23. y = (–2) x

24. Explain why the graphs of #21-23 are not exponential functions. What in the

equations is wrong?

Answer the following multiple choice questions based on your knowledge of exponential functions and their graphs. Pay attention to increasing and decreasing equations. 25. If the equation of y = 5 x is graphed, which of the following values of x would produce a point closest to the x-axis?

a. 0 b. –1 c. 23

d. 74

26. If the equation of y = 12

x

is graphed, which of the following values of x would

produce a point closest to the x-axis?

a. 14

b. 34

c. 53

d. 83

27. If the equation of y = 13

x

is graphed, which of the following values of x would

produce a point closest to the x-axis?

a. 0 b. –1 c. 23

d. 74

28. Which multiple choice ordered pair represents the y-intercept for the function y = 2 x ?

a. (0,0) b. (0, 1) c. (0, 2) d. there is no y-intercept 29. Select the correct multiple choice response. The graph of y = 5 x lies in which quadrants?

a. Quadrants 1 and 2 b. Quadrants 1 and 3 c. Quadrants 1 and 4

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30. Select the correct multiple choice response.

The graph of y = 110

x

contains which of these points?

a. (0, 0) b. (0, 10) c. (0, 1) d. (0, 110

)

31. Which multiple choice ordered pair represents the x-intercept for the function y = 4 x ?

a. (0, 0) b. (0, 1) c. (1, 0) d. there is no x-intercept

32. Use the graph of y = 2x to answer the following multiple choice question. If the equation y = 2x is graphed, which of the following values of x would produce a point closest to the x-axis?

a. 14

b. 34

c. 53

d. 83

33. Given the expression x n where x > 1 and n > 1, which multiple choice statement is true? a. the value of x n = 0 b. the value of x n > 0 c. the value of x n < 0

d. the value of x n = 1

34. Given the expression x n where x > 1 and n = 0, which multiple choice statement is true? a. the value of x n = 0 b. the value of x n > 0 c. the value of x n < 0

d. the value of x n = 1

35. Given the equation y = x n where 0 < x < 1 and n > 1, which multiple choice statement is true? a. y = 0 b. y > 0 c. y < 0 d. y = 1

36. Given the equation y = x n where x > 1 and n < 0, which multiple choice statement is true? a. y = 0 b. y > 0 c. y < 0 d. y = 1

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Algebra 2 Unit 8 Worksheet 2 CALCULATORS ARE NOT ALLOWED

Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes. Examples of such phenomena include the studies of populations, bacteria, the AIDS virus, radioactive substances, electricity, temperatures and credit payments, to mention a few.

Any quantity that grows or decays by a fixed percent at regular intervals is said to possess exponential growth or exponential decay.

Such a situation is called Exponential Decay.

Such a situation is called Exponential Growth.

The time required for a substance to decay and fall to one half of its initial value is called the half-life. Radio-isotopes of different elements have different half-lives. Some people are frightened of certain medical tests because the tests involve the injection of radioactive materials. Doctors use isotopes whose radiation is extremely low-energy, so the danger of mutation is very low. The half-life is long enough that the doctors have time to take pictures, but not so long as to pose health problems. They use elements that are not readily absorbed by the body but are voided or flushed long before they get a change to decay within your body.

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For the following word problems we will be using the exponential equation y = A ( )thb .

Where A is the initial amount b is the amount of growth (or decay) that occurs in h time t is time 1. Technetium-99m is one of the most commonly used radioisotopes for medical purposes. It has a half life of 6 hours. If 0.5 cc’s (which is less than a teaspoon) of Technetium-99m is injected for a scan of a gallbladder, how much radioactive material will remain after 24 hours?

Use the formula y = A t61

2

where A = the number of cc’s present initially

t = time in hours 2. When a plant or animal dies, it stops acquiring Carbon-14 from the atmosphere. Carbon-14 decays over time with a half-life of 5730 years. How much of a 10mg sample will remain after 11,460 years?

Use the formula N = N0 th1

2

where N0 is the initial amount

N = the amount remaining t = time in years h = half life 3. One certain element has a half-life of 1600 years. If 300 grams were present

originally, how many grams will remain after 3200 years?

4. The radioactive gas radon has a half-life of 3 days. How much of an 80 gram sample will remain after 9 days?

5. The radioactive gas radon has a half-life of approximately 132

days. About how

much of a 200 gram sample will remain after 1 week?

6. The population of a certain country doubles in size every 60 years. The population is now 1 million people. Find its size in 180 years.

y = A 60(2)t

A = initial population t = time elapsed in years

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7. Bacteria populations tend to have exponential growth rather than decay. Suppose a certain bacteria population doubles in size every 12 hours. If you start with 100 bacteria how many will there be in 48 hours? 8. A certain population of bacteria doubles every 3 weeks. The number of bacteria now is only 10. How many will there be in 15 weeks? 9. A culture of yeast doubles in size every 20 minutes. The size of the culture is now 70. Find its size in 1 hour (remember to convert 1 hour to minutes.) 10. The growth of a town doubles every year. If there are 64,000 people after 4

years, find the initial population.

11. The number of people with a flu virus is growing exponentially with time as shown in the table below. Flu Virus Growth

Day Number of People 0 400 1 800 2 1600

Which multiple choice equation expresses the number of bacteria, N, present at any time, x ?

a. N = 400 x b. N = 400 + 2x c. N = 800 • 2 –x

d. N = 400 • 2x

12. In the early years of the century the national debt was growing exponentially with time as shown in the table below. National Debt

Year Debt 0 30,000 1 60,000 2 120,000

Which multiple choice formula expresses the debt, y, at any time t ? a. y = 30,000 •2t b. y = 10000 • 3 t

c. y = 3 • 10t d. y = 30,000 + 2t 13. An epidemic of bubonic plague grew exponentially by the formula A = A0 • 2t where A0 = original amount infected t = time passed in weeks If 512,000 people were infected after 8 weeks, find the original amount that were infected.

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Use estimation for the following multiple choice questions: Choose the best multiple choice response for the following:

14. 943

a. 1.3 b. 3.9 c. 11.8 d. 35.5

15. ( ) ( )3212 2•

a. 16.9 b. 33.9 c. 67.3 d. 117.5

16. A radioactive element decays over time according to the equation: y = A 3001

2

t

If 1000 grams were present initially, how may grams will remain after 650 years? a. 444 b. 222 c. 111 d. 55.5

17. Boogonium decays using the formula: A = I • t

h2−

The half life of Boogonium is 4 hours. How much of a 24 gram sample will remain after 6 hours. Choose the best multiple choice response. a. 0.4 b. 3.2 c. 8.5 d. 16.9

18. Geekonium-25 decays using the formula: A = I • t

h2−

The half life of Geekonium-25 is 2 years. Find how much of a 160 gram sample remains after 8 years.

Unit 8 Worksheet 3

Determine the exponent needed to change the left number into the right number. You may use positive, negative, zero, and fractional exponents. Guess and Check: 1. 5 → 25 2. 4 → 64 3. 2 → ½ 4. 3 → 1/9 5. 6 → 1 6. 27 → 3 7. 5 → 1/125 8. 16 → 4 9. 8 → 22

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Logarithms (or logs) are used to find the exponents to help us solve exponential equations. Structure of a logarithm: yblog = x

b is the base y is the value

x is the exponent on b to yield y

Example: Simplify 2log 8= ? =3 (because 23 = 8) Simplify #10-29.

10. log6 36 11. log2 16 12. log10 100 13. 31log9

14. 2log 2 2 15. log7 1 16. log5 125 17. log4 16

18. log3 81 19. log6 6 20. log3 1 21. log8 4

22. 51log

25

23. 21log8

24. 6log 6 6 25. 5log 25 5

26. 37log 49 27. 5

3log 9 28. 32

1log4

29. 101log

100

Logarithms with a base 10 are called common logarithms. The base of 10 is implied and not shown. For example, log 1000 is equivalent to log 10 1000 Simplify: (Remember, when no base is given it is assumed to be base 10)

30. log 100 31. 1log

10

32. log 1 33. log 10

34. log 0.01 35. log 10 3 36. log 0.0001 37. log 100

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Unit 8 Worksheet 4 Log Rules : (b and y must be positive numbers, b ≠ 1 ) logb y x= xb y= log b xb = x log y

b b = y log 1b = 0 3.14π ≈ 2.718e ≈ Remember, if no base is shown assume it is base 10, the common log. log y x= 10log y x= 10 x = y If base e is used it is called a natural log. Instead of writing log we use ln ln y x= =loge y x xe = y (Remember, e is just an irrational number. It is approximately 2.718; see Page 492 in your textbook)

Restrictions: You can’t take log 0 or log (of a negative number)

With bases, you can’t do log 0 base or log 1 base or log negative base

Verify the log by rewriting the equation into exponential form.

1. log2 32 = 5 2. log3 9 = 2 3. log7 7 = 12

4. log3 181

= – 4

Rewrite the equation in logarithmic form.

5. 43 = 64 6. 329 = 27 7. .210 0 01− = 8.

34 116

8−=

Simplify:

9. 5log 235 10. 72log 2 11. log910 12. 12log 1

13. 7log 49 14. 8log 64x 15. 4log 16x 16. 2log 16x

17. 8log 1 18. 57log 7 19. 3log 113 20. 6log 6

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Write each equation in exponential form. 21. ln 8 = 2.08 22. ln 100 = 4.61 23. log 9.86 2π =

24. log1000 3= 25. ln 1097 = 7 To graph a log equation: 1. First rewrite it in exponential form 2. Make a table of values. Look at the equation and see which letter (x or y) is the exponent and put the numbers 2, 1, 0, –1, –2 in that column. 3. Plot the points and connect with a curve

Graph #26-29 on graph paper. Be sure to show the table of values and the

exponential equation.

26. y = 2log x 27. y = 5log x 28. y = 14

log x

29. Graph y = 3 x and y = 3log x on the same grid.

Choose the correct multiple choice.

30. Which is equivalent to 1216 = 4 ?

a.

4

1log

2= 16 b. 16

1log2

= 4 c. 161log 42

= d. log 4 16 = 12

31. Which is equivalent to logm n p= ?

a. m n = p b. m p = n c. n p = m d. p n = m 32. Which is equivalent to log k = w ?

a. 10 w = k b. 1 w = k c. k w = 10 d. 10 k = w 33. Given: y = 5 x which statement is true?

a. y > 0 for all values of x b. y > 0 for all values of x

c. y < 0 for all values of x d. y < 0 for all values of x

34. When is the following statement true? 7log7 x = x

a. for all values of x b. for some values of x c. for no values of x

d. can’t determine

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35. In the equation logx y z= which statement is true about the value of z ?

a. z must always be positive b. z can never equal 0 c. z can never equal 1

d. there are no restrictions on z 36. When is the equation 6log 6y = y ?

a. for all values of y b. for no values of y c. for some values of y d. cannot determine 37. Which expression is equivalent to ln x = y ?

a. 10 y = x b. ey = x c. xy = e d. ey = x 38. Which expression is equivalent to 6log 36x ?

a. 2x b. 36x c. 2 x d. 6 x 39. Which expression is equivalent to log 1000 x ?

a. 1000x b. x3 c. 10x d. 3x

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Unit 8 Worksheet 5 On pg. 507 in our text are the Laws of Logarithms 1. Multiplication Property: logb MN = logb M + logb N

2. Quotient Property: logbMN

= logb M − logb N

3. Power to a Power Property: log nb M = n logb M

If you are given: 10log 4 = .6021 and 10log 6 = .7782 , use the Laws and the given to find the following. Justify each step with the properties listed above or basic operations property.

Example 10log 24

10log (4 6)⋅ Factors of 24

10log 4 + 10log 6 Multiplication Property

.6021 + .7782 Substitution Property

1.3803 Addition

2. 10log 16 3. 103log2

4. 101log4

5. 10log 36

6. 10log 6 7. 10log 2 (hint: 2 = 4 ) 8. 101log ( )

16

Even though we were only given 10log 4 and 10log 6 we know 10log 10 = 1 and 10log 100 = 2 9. 10log 40 10. 10log 400

In the preceding problems we had to work with decimal values. The following problems involve the same 3 laws of logarithms, but we will use variables instead of decimals. Given: 2log 9 = c and 2log 10 = d

Find the following in terms of c and d

11. 2log 90 = 12. 2log 81 = 13. 210log ( )9

=

14. 2log 10 = 15. 21log ( )9

= 16. 21log ( )

10

17. 2log 3 18. 2log 900 = 19. 3

2log ( 9) = 20. You were given the 2log 9 and 2log 10 , but you also know 2log 2 = 1, use this to find

2log 18 =

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Select the correct multiple choice:

21. log xy2 =

a) 2 log xy b) 2 log x + log y c) 2 log x + 2 log y d) log x + 2 log y

22. log x • log y = a) log (x + y) b) log (x • y) c) log x + log y d) none of these

23. log x – log y =

a) log xy

b) logxlogy

c) both ‘a’ and ‘b’ d) neither ‘a’ or ‘b’

24. log 1004x =

a) 4x b) 6x c) 8x d) 16x

25. log 2x =

a. log 2 + log x b. log 2 • log x c. 2 + x d. 2x

26. log 3 =

a. log ( 12

• 3 ) b. log 32

c. 12

log 3 d. 12

log 3

27. log x + log y + log z =

a. log (x + y + z) b. log (x • y • z) c. log x • log y • log z

28. log x (x w ) =

a. log w b. log x w c. w d. x w

29. Which student solved for x correctly in the following problem? 2 log x = 4

Alice Bob Carl David

2 log x = 4 2 log x = 4 2 log x = 4 2 log x = 4

log x2 = 4 log x2 = 4 log x2 = 4 log x2 = 4

x2 = 4 x2 = 4 x2 = 104 x2 = 104

x = 2 x = ± 2 x2 = 10000 x2 = 10000

x = 100 x = ± 100

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30. Which student solved for x correctly in the following problem? 2 log 3 + log x = log 36

Astro Bella

2 log 3 + log x = log 36 2 log 3 + log x = log 36

log 9 + log x = log 36 log 9 + log x = log 36

log 9x = log 36 log (9 + x) = log 36

9x = 36 9 + x = 36

x = 4 x = 27

Chu Domingo 2 log 3 + log x = log 36 2 log 3 + log x = log 36

2(log 3 + log x) = log 36 2(log 3 + log x) = log 36

2 log 3x = log 36 2 log 3x = log 36

log 3x2 = log 36 log (3x)2 = log 36

3x2 = 36 9x2 = 36

x2 = 12 x2 = 4

x = 12 x = 2

Unit 8 Worksheet 6 A. If we write 2log 10 in exponential form we get x2 10= We are going to have to

approximate the value of this. We know 32 = 8 x2 = 10 42 = 16 So the exponent, x, will be between the consecutive integers 3 and 4. B. 3log 25 becomes x3 25= Between what 2 consecutive integers will x lie?

3 2 = 9 3 x = 25 3 3 = 27 So x is between 2 and 3. Would it be closer to 2 or closer to 3? ______

Determine which two integers the following logarithms lie between:

1. 2log 30 2. 7log 9 3. 4log 100

4. 3log 200 5. 10log 7500 You can convert all logarithm problems to equivalent logarithms with base 10 or e. Below is the formula to convert logarithms to any base. Change of Base

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log c a is currently in base ‘c’. To change it, write it as a fraction

log c a = log alog c

You’ll notice that no base was given. You can use any base. For example:

log c a = log alog c

= 6

6

log alog c

or 4

4

log alog c

or 8

8

log alog c

Change of Base Formula

log c a = b

b

log alog c

(where ‘b’ can be any positive base ≠ 1)

Since most calculators only work in base 10 or base e, it is best to change to one of them.

log c a = 10

10

log alog c

or ln aln c

Rewrite the following using the change of base formula. Change into the indicated base. 6. log 5 7 to base 2 7. log 9 4 to base 6

8. log 2 3 to base 10 9. log 8 5 to base e

You can use the change of base formula in reverse.

If you are given b

b

log alog c

you can condense it to a single log by dropping the base b.

b

b

log alog c

= log c a

Express the following as a single log:

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10. 5

5

log 8log 7

11. 9

9

log 12log 4

12. 2

2

log 6log 10

13. log 11log 5

14. ln4ln3

Express the following as a single log. Then simplify the final answer.

15. 4

4

log 49log 7

16. 8

8

log 81log 3

17. log64log 4

18. 5

5

log 2log 8

19. log 2log 2

20. ln32ln2

21. log 5 7 =

a. log 5 – log 7 b. log 7 – log 5 c. 7 • log 5 d. log7log5

22. 8log 20 =

a. 3

3

log 20log 8

b. 20log8

c. log 20 – log 8 d. 20 log 8

23. 7

7

log 16log 8

=

a. log 716 – log 7 8 b. log 8 16 c. log 2 d. 2

Algebra 2 Unit 8 Worksheet 7 Solve for x using common bases.

1. 3 x = 127

2. 28 x+ = 2 3. 14 x− = 8

4. 2 127 x− = 3 5. 3 5 14 16+ +=x x 6. ( 5) 43 9x x− + =

7. 2 625 5x x+= 8. 1 16 36x x+ −= 9. 1 410 100x x− −=

10. 5 125x = 11. 249 7 7x− = 12. 6 36 6x =

Solve for x using inverse properties of exponents.

13. 13 5x = 14.

32 8x = 15.

52 32x =

16. 344 108x = 17.

143 6x = 18.

325 40x

−=

19. 53( 5) 2 30x + − = 20.

12( 1) 10x − =

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Unit 8 Worksheet 8 Solve for x. Some problems may have no solution. 1. log 2 x = 3 2. log 2 x = – 4 3. log 5 x = 3

4. log 2 (–2) = x 5. log x 144 = 2 6. 5log 235 = x

7. log 4 x = 12

8. log 8 x = 23

9. log 8 1 = x

10. log 1 6 = x 11. log 6 6 3 = x 12. log 4 x = 32

−

13. log x 27 = 32

14. log 7 (–49) = x 15. ( 9)log x− = 12

16. log16 x = – 12

17. log 7 0 = x 18. log 5 0 = x

19. 19

1log2

x = − 20. log x 8 = – 1 21. log x 16 = 2

22. 3log (27 3) = x 23. log 10 5 = x 24. log x 8 = 34

25. log 5 (25 3) = x 26. log 2 (4 5) = x 27. log 2 7x = log 2 98

28. 3 log 5 4 = log 5 2x 29. log 7 4x = log 7 5 30. 2 ln 9 = ln 3x

31. 7 7log 2 log 16x = 32. log 5 (2x + 12) = log 5 (3x + 4)

33. 2 log 8 x = log 8 100 34. 28 8log 3 log 81x =

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Algebra 2 Unit 8 Worksheet 9 Solve for x using properties of logs. On problems involving π or e leave answers in

terms of π or e. Do not approximate. Some problems will have no solution.

1. log 7 x = log 7 2 + log 7 3 2. log 6 x = 2 log 6 3 + log 6 5

3. log 5 (x + 3) = log 5 8 – log 5 2 4. log x – log (x – 5) = log 6

5. ln (3x + 5) – ln (x – 5) = ln 8 6. log 11 x = 32

log 11 9 + log 11 2

7. 2log 5 log 125x = 8. log 6 9 + log 6 x = 2

9. log x + log 25 = 3 10. log 2 52 – log 2 x = 2

11. 2 log 6 2 + log 6 18x = 3 12. ln 4 ln 8x =

13. log xπ = 3 14. log 5 log 7xπ π+ =

15. 64log 32 x= 16. log 6 x + log 6 (x – 5) = 2

17. 2 log 4 x = 3 18. ln x = 2

19. ln x + ln 5 = 4 20. ln x – ln 6 = 2

21. log 2 4x – log 2 (x – 1) = 3 22. log 2 x + log 2 (x – 6) = 4

23. 2 log 2 + log x = 2 24. 2 ln 7 + ln x = 4

25. log 20 + log 5 = x 26. log 6 9 + log 6 4 = x

27. log 5 (2x – 7) = 0 28. ln (x – 9) = 1

29. Identify which step has the error in the solution of 2 log 7 x = log 7 2 + log 7 50

Step 1: 2 log 7 x = log 7 (2 • 50)

Step 2: 2 log 7 x = log 7 100

Step 3: log 7 x = 7100log2

Step 4: log 7 x = log 7 50

Step 5: x = 50

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30. Which line has an error in it?

log 6 6 + log 6 6 = x

1. log 6 6 6 = x

2. 6 6 6x =

3. 1

1 26 6 6x = •

4. 126 6x =

5. x = 12

31. What multiple choice helps when solving 2 x = 32 ?

a. 32 ÷ 2 = 16 b. 2 • 32 = 64 c. 32 = 25 d. 21 = 2 32. What multiple choice helps when solving log 5 x + log 5 4 = log 5 24

a. log x + log y = log (x + y) b. log x + log y = log (xy)

c. p log x = log x p d. log x – log y = log xy

33. What multiple choice helps when solving ln x = 4

a. ln x = ln e x b. e ≈ 2.718 c. 4 1 = 4 d. 4 0 = 1

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Unit 8 Worksheet 10 CALCULATORS ARE NOT ALLOWED

If we are given =2 2log 2 log 5x , how would we solve for the exponent, x? We use logarithms to help us solve these exponential functions. Equation: =2 5x =2 2log 2 log 5x x = 2log 5 (our calculator could give us a decimal approximation, but for now this is how we write our answers) Solve the following problems for x by introducing logs. Leave answers in log form. 1. x7 12= 2. x5 30= 3. x10 92=

4. 2x8 74= 5. x 34 22+ = 6. xe 43=

Choose the correct multiple choice response: 7. x7 14=

a. x = 2 b. x = log 14 c. x = log14log7

d. x = log 2

8. If x = 4log 15 which is true about x?

a. x < 0 b. 0 < x < 1 c. 1 < x < 2 d. x > 2 9. 10 x = 200

a. x = log 200 b. x = 200log 10 c. x = 20 d. x = 10

10. e x = 4

a. x = log 4 b. x = ln 4 c. x = ln e 4 d. x = 4

11. 2 x + 1 = 13

a. x = 2log 13 1− b. x = 6 c. x = log12log2

d. x = log 6

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12. Which step has the error: ln 8 + ln x = 5

Step 1 ln 8x = 5

Step 2 8x = 10 5

Step 3 8x = 100,000

Step 4 x = 100,000

8

13. Which step has the error: x 17 9+ =

Step 1 x 17log 7 + = 7log 9

Step 2 x + 1 = log 7 9

Step 3 x = log 7 9 – 1

Step 4 x = log 7 8

Algebra 2 Unit 8 Review 1 CALCULATORS ARE NOT ALLOWED

Simplify:

1. 13125 2.

12100

− 3.

3416

− 4.

351

32

−

Write the following in logarithmic form. 5. 4 3 = 64 6. 110 0.1− = 7. 1 2.718e = 8. ba c=

Write the following in exponential form.

9. log 2 16 = 4 10. 51log 225

= −

11. log 1000 = 3

12. ln 148 = 5 13. log 7 1 = 0 14. log 31 3π =

Simplify. Some problems will have no answer. 15. 5log 85 16. log 4 64 17. 2ln ( )e

18. log 5 0 19. 71log49

20. ln (1)

21. 1lne

22. 2

log 8 23. 38log ( 2)

- 23 -

24. 13

log 9

25. 91log

3−

26. log 5 1

27. 1log10

28. 7ln ( )e 29. 27log 7 x 30. 4

6log 36 x

Solve for x. On problems involving π or e leave answers in terms of π or e . (Do not approximate.) Some problems will have no solution. Some problems will have answers in log terms.

31. 3 4x = 3 3 – x 32. 4 x = 23 33. 2 x – 3 = 1

16

34. 6 x = 11 35. 5 x = 125 36. 32 64x =

37. 122 6x = 38. 29 x = 17 39.

132(7 1) 4 0x− − =

40. 23xe = 41. 9 2x = 27 x – 1 42. 1 30xe + =

43. log 5 x = – 3 44. 1log 12x

= −

45. log x 125 = 34

46. ln x = 7 47. log 5 (– 5) = x 48. 3log π π = x

49. 4

1lne

= x 50. ln x = – 2

51. log 6 4 + log 6 x = 2 52. log 6 4 + log 6 x = log 6 12

53. log 7 (x + 3) – log 7 x = log 7 2 54. ln (x) + ln (3) = ln (x + 4)

55. log x + log (x – 3) = 1 56. 2 log 6 x + log 6 3 = log 6 75

Express as a single log and simplify, if possible. 57. log 5 10 + log 5 4 58. log 672 – log 6 2 59. 2 log 210 – log 2 25

60. 7

7

log 11log 4

61. log 50 + log 4 – log 2 62. 1 1log 27 log93 2

−

63. 6

6

log 8log 2

64. ln18ln 5

65. log32log5

Given: log 2 = .3010 log 6 = .7781 Find the following:

66. log 12 67. log 3 68. log 4 69. log 12

70. log 13

(this is equivalent to log 26

) 71. log 20

72. log 72

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Given: log 3 = k log 5 = f Find the following:

73. log 15 74. log 15

75. log 50 76. log 45

Graph the following:

77. y = 7 x 78. y = 13

x

79. y = log 4 x

Answer individual questions: 80. Between what 2 consecutive integers does log 1230 lie?

81. If the equation y = 4 x is graphed, which of the following multiple choice values of x would produce a point closest to the x-axis?

a. 14

b. 0 c. – 2 d. 3

82. A radioactive substance decays by the given formula. How much of a 160 gram sample will remain after 6 hours?

y = A 31

2

t

A = initial amount

t = time in hours

83. A radioactive element decays over time as shown in the table below. Which multiple choice equation expresses the amount of grams, y, present at

a. y = 12

g b. y = 100 12

h

hour, h?

c. y = 12

h g• • d. y = 100 61

2

h

84. Given the equation y = log x, which multiple choice statement is valid?

a. x < 0 b. x < 0 c. x = 0 d. x > 0

85. Which multiple choice is equivalent to log 20 – log 5

a. log 4 b. log 15 c. log 20log5

d. log 100

86. Which multiple choice is equivalent to log 6 24 ?

a. log 4 b. 3

3

log 24log 6

c. log 6 11 + log 6 13 d. (log 6 2)(log 612)

hour grams 0 100 6 50 12 25

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87. Which multiple choice is the solution to the equation 9x = 45 ?

a. x = log 45log9

b. x = 5 c. x = log 5 d. x = log 45 – log 9

88. Given the expression xn where x > 1 and n > 1, which multiple choice statement is true?

a. the value of xn = 0 b. the value of xn > 0

c. the value of xn < 0 d. the value of xn = 1

89. Given the expression xn where x > 1 and n =0, which multiple choice statement is true?

a. the value of xn = 0 b. the value of xn > 0

c. the value of xn < 0 d. the value of xn = 1

90. Given the equation y = xn where 0 < x < 1 and n > 1, which multiple choice statement is true?

a. y = 0 b. 0 < y < 1 c. y < 0 d. y = 1

Algebra 2 Unit 8 Review 2 CALCULATORS ARE NOT ALLOWED

Choose the correct multiple choice response in # 1 – 22. 1. Write 37 343= in logarithmic form. a) 7log 343 3= b) 3log 343 7= c) 7log 3 343= d) 3log 7 343= 2. Write 10log 0.0001 4= − in exponential form a) 40.0001 10− = b) 104 0.0001− = c) 410 0.0001− = d) 0.000110 4= −

3. Evaluate 16log 4 a) 2 b) −2 c) 12

d) 12− e) 1

4

4. Solve for x: log 9 2x = a) 3 b) 4.5 c) −3, 3 d) 81 e) −3

5. Solve for x: 5log 3x = − a) −15 b) −125 c) 1125

d) 1125− e) 3 5

6. Evaluate: 65log 5 a) 5 b) 6 c) 25 d) 36 e) none of these

7. Evaluate: 7log 497 a) 7 b) 2 c) 1 d) 49 e) none of these

8. Solve: 2log ( 8) x− = a) 3 b) −3 c) 13

d) 13− e) none of these

9. Solve: 2 21log log 1253

y = a) 5 b) 2 c) 375 d) 1253

e) none of these

10. Solve: 2 2 21 log 4log 2 log 42

x = − a) 2 b) 4 c) 16 d) 32 e) none of these

11. Solve: 4 4log ( 1) log ( 1) 2m m− + − = a) 3 b) 5 c) 9 d) −3, 5 e) none of these 12. Solve: log 2 (x) + log 2 (x + 2) = 3 a) 2 b) 4 c) - 4 d) 2, – 4 e) 2, 4 13. Solve: log x 9 = 2 a) 3 b) - 3 c) 3, -3 d) 3 e) none of these

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14. Given: log 2 c= and log 7 d= , Find: log 56 a) c 3 + d b) c3d c) c 3 + d d) 3c + d e) none of these

15. Solve for x: 1log ( ) 38x = − a) 1

2 b) 2 c) 1

2− d) −2 e) none of these

16. Solve for x: 5log 0 x= a) 0 b) 1 c) 5 d) −1 e) none of these

17. Solve for x: 5 5 512 log 6 log 27 log3

x− = a) 33 b) 2 c) 3 d) 4 e) none of these

18. If 4log 7 = n, find 41log ( )7

a) 1n

b) 1n− c) − n d) 1 − n e) none of these

19. If log 2 c= and log3 = d, find log 6

a) 1 12 2

c d+ b) 12

c d+ c) cd d) c d+ e) 12cd f) none of these

20. Which of the following is true about the graph of logy x= ? a) it passes through (0,1) b) it lies in quadrants 1 and 2 only

c) it is a decreasing graph d) the value of x will never be 0 21. Solve for x: 73log 4 + 72 log 2 = 7log x a) 64 b) 16 c) 256 d) 32

22. Find the value of x: 1254 log3

x=

a) 50 b) 5 c) 25 d) 625 True or False 23. log10 x x= 24. ln xe x= 25. 3 7log 9 log 49= 26. 22 x = 4x

27. 3 32 x+ = 18 x+ 28. 122x = 2x

Simplify:

29. log 5 + 2 log 4 – 3 log 2 30. 7

7

log 32log 2

31. log 6 3 + log 6 12

Solve for x:

32. 12

x

= 34 33. log 6 (x + 5) + log 6 x = 2 34. log 6 3 + log 6 x = log 62

35. 81log

3x −= 36. 3log (27 3) = x 37.

12 1

8x =

38. 1 12 8

x =

39. ln x = 4 40. log 3 (– 9) = x

- 27 -

41. Graph: y = 15

x

42. Graph: y = 2log x

43. Express as a single log: 2 ln 7 + ln 2 44. Express as a single log and simplify: 2 log 6 2 + 2 log 6 3 45. If log 5 2 = k and log 5 3 = m find log 5 30 in terms of k and m 46. Given the equation y = xn where x > 1 and n < 0, which multiple choice statement is true? a) y = 0 b) y > 0 c) y < 0 d) y = 1 47. If the equation y = 4x is graphed, which value of x would produce a point closest to the x axis?

a) 3 b) – 5 c) 23

− d) 13

48. If the equation, 13

x

y =

is graphed, which value of x would produce a point closest to the

x-axis? a) 7 b) 0 c) 2 d) – 6

49. If the equation, 97

x

y =

is graphed, which of the following values of x would produce a point

closest to the x axis?

a) 35

− b) 47

c) 13

d) 23

−

50. If the equation, x = log2y is graphed, which of the following values of x would produce a point farthest from the x axis? a) – 8 b) 2 c) – 3 d) 9 Simplify 51. log327x 52. log416x 53. log82x

Unit 8 Review #3

1. Given the equation ny x= where 1x > and 0n > , which statement is true? a) y = 0 b) y < 0 c) y > 1 d) 0 < y < 1 e) y is undefined 2. Given the equation ny x= where 0 1x< < and 0n > , which statement is true? a) y = 0 b) y < 0 c) y > 1 d) 0 < y < 1 e) y is undefined 3. Given the equation ny x= where 0 1x< < and 0n < , which statement is true?

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a) y = 0 b) y < 0 c) y > 1 d) 0 < y < 1 e) y is undefined 4. Bacteria are growing exponentially with time as shown in the table below. Write the equation that expresses the number of bacteria, y, present at any time, t ? Bacteria Growth

Hour Bacteria 0 5 1 10 2 20

5. Bacteria are decaying exponentially with time as shown in the table below. Write the equation that expresses the number of bacteria, y, present at any time, t ? Bacteria Growth

Hour Bacteria 0 100 1 50 2 25

Simplify the following:

6. ( )3log 9− 7. ( 3)log 9− 8. 31log9

9. 3log 9x 10. 5log 125x Approximate the following: 11. 2log 9 12. 4log 3 13. 3log 30