* objectives: * use the properties of exponents. * evaluate and simplify expressions containing...

*Objectives:

*Use the properties of exponents.

*Evaluate and simplify expressions containing rational exponents.

*Solve equations containing rational exponents.

*A number is written in scientific notation

when it is in the form a x 10n where a is

between 1 and 10 and n is an integer.

At their closest points, Mars and Earth are approximately 7.5 x 107 kilometers apart.

a. Write this distance in standard form.

b. How many times further is the distance Mars Pathfinder traveled than the minimum distance

between Earth and Mars? (the pathfinder traveled 4.013 x 108)

75,000,000

401,300,000 75,000,000

= .535 x 101 = 5.35

Please work with your table

partner to go through the

graphing calculator exploration.

**If we have time at the end**

a. 1251/3 =

b. (s2t3)5 =

c.

d.

e. (81c4)1/4 =

f. 6253/4 =

g. Simplify:

h. Solve: 734 = x3/4 + 5

1

52

34

3

x

yx

3257 tsr

25

1510ts

5

9xy

c3

125

23

225

27

tsr rsttsr 123

6561x

PropertyProperty DefinitionDefinition

ProductProduct

Power of a PowerPower of a Power

Power of a QuotientPower of a Quotient

Power of a ProductPower of a Product

m

mm

ba

ba

)(

mmm baab )(

nmmn aa )(

mnmn aaa

PropertyProperty DefinitionDefinition

QuotientQuotient

Rational ExponentRational Exponent

Negative ExponentNegative Exponent

Zero ExponentZero Exponent

n mnm aa /

10 a

nn

aa

1

nmn

m

aaa

WARM-UP:

*Learning Targets:

*I can solve problems involving exponential growth and decay.

*I can evaluate expressions involving logarithms.

*I can solve equations and inequalities involving logarithms.

*An exponential function is of the form y = bx where the base b is a positive real number and the exponent is a variable.

Please work with your table

partner to go through the

graphing calculator exploration.

b > 1b > 1 0 < b < 10 < b < 1

DomainDomain

RangeRange

y- intercepty- intercept

BehaviorBehavior

Horizontal AsymptoteHorizontal Asymptote

Vertical AsymptoteVertical Asymptote

0y

)(lim xfx

All real numbers All real numbers

(0, 1) (0, 1)

0y

0)(lim

xfx

none none

Y=0 Y=0

What happens when b=1?

xy blog xb y

Write each expression in exponential form:

a. b.

3

225log125

3

12log8

2512532

2831

Write each equation in logarithmic form:

a. b. 6443

27

13 3

364log4 327

1log3

*Evaluate the expression

y49

1log7

49

17 y

*How long would it take for 256,000 grams of Thorium-234, with a half-life of 25 days, to decay to 1000 grams?

x)2

1(2560001000

x)2

1(

256000

1000

))2

1log(()

256

1log( x

)2

1log()

256

1log( x

x)21

log(

)2561

log(x)2

1(

256

1

8xyears200

PropertyProperty DefinitionDefinition

ProductProduct

QuotientQuotient

PowerPower

EqualityEquality

nmmn loglog)log(

nmn

mloglog)log(

mpm p loglog

nmthennmIf ,loglog

“Drop it down”

“lop it off”

Solve each equation (Warm-Up):

a.

b.

c. 6log)1(loglog 111111 xx

2

164log 3

1

p

)45(log)112(log 44 xx

Solve each equation:

a.

2

164log 3

1

p

2

14log p

????42

1

p

You don’t HAVE to do anything fancy for this one….just your brain…..ummmmmm

16p

4p

Solve each equation:

b.

)45(log)112(log 44 xx

45112 xx

x315

x5

Solve each equation:

c.

6log)1(loglog 111111 xx

6)1( xx

6log)]1([log 1111 xx

062 xx

62 xx 0)2)(3( xx

2,3x

6)1(loglog 22 xx

6log 97 x

6)1(log 39 x

d.

e.

f.

6)1(loglog 22 xxd.

6)]1([log2 xx

xx 262

xx 262

640 2 xx

6log 97 x

967 x

x9 117649

9117649 x

x659.3

e.

6)1(log 39 xf.

6)1(loglog 1111 xx

Section 11.2 & 4: Exponential and Logarithmic Functions (Graphing) 

Objectives:Graph exponential functions and inequalities.Graph logarithmic functions and inequalities.

*Graph the exponential functions y=4x, y=4x+2, and y=4x-3 on the same set of axes. Compare and contrast the graphs.

Graph the exponential functions y= , y= , and y=

on the same set of axes. Compare and contrast the graphs.

x

5

12

x

5

16

x

5

1

*Graph y < 2x + 1.

xy 3

)1(log3 xy

2log5 xy

*Solve: 32x-1 = 4x

*Solve 2log8 x – log8 3 = log8 10

)4(log2 xy

According to Newton’s Law of Cooling, the difference between the temperature of an object and its surroundings decreases in time exponentially. Suppose a certain cup of coffee is 95oC and it is in a room that is 20oC. The cooling for this particular cup can be modeled by the equation y = 75(0.875)t where y is the temperature difference (from the room) and t is the time in minutes.

*Find the temperature of the coffee after 15 minutes.

15)875(.75y oy 12.10 oo 12.3012.1020

The equation A = ,where A is the final amount,

P is the initial (principal) amount, r is the annual interest

rate, n is the number of times interest is paid/compounded

per year, and t is the number of time periods, is used for

modeling compound interest.

*Suppose that a researcher estimates that the initial population of varroa (the parasite that kills bee populations) in a colony is 500. They are increasing at a rate of 14% per week. What is the expected population in 22 weeks?

*Determine the amount of money in a money market account providing an annual rate of 5% compounded daily if Marcus invested $2000 and left it in the account for 7 years.

*How long would it take for 256,000 grams of Thorium-234, with a half-life of 25 days, to decay to 1000 grams?

x)2

1(2560001000

x)2

1(

256000

1000

))2

1log(()

256

1log( x

)2

1log()

256

1log( x

x)21

log(

)2561

log(x)2

1(

256

1

8xyears200

1. Graph y = log4 (x+2) – 1

2. Solve 103x – 1 = 6 2x

3. If I want to make $500 in 2 years, and I have $2000 to invest now, what interest rate do I need to convince my bank to give me if they add interest semi-annually?

*Objectives

*Find common logarithms of numbers

*Solve equations and inequalities using common logarithms

*Solve real-world application with common logarithmic functions.

*A common logarithm is a logarithm with a base of 10

*Given that evaluate each logarithm.

a.log 7,000,000

b.b. log 0.0007

8451.07log

85.6

15.3

1000000log7log 68451.

Evaluate each expression

a. log 5(2)3

b.

6

19log

2

=

= 6log19log2

40log 602.1

779.1

*Graph y > log (x – 4).10

*Graph y = log x + 2

*Graph y = log (x + 1) - 4

*Graph y = -log (x) + 5

*Graph y = 3log (x – 4)

If a, b, and n are positive numbers and neither a nor b is 1, then the following equation is true:

a

nn

b

ba log

loglog

*Find the value of using the change of

base formula

1043log9

9log

1043log 163.3

* Solve the following equations:

a. 63x = 81

b. 54x = 73

8175.x

667.x

132.. xc

*WARM-UP

*Solve using change of base.

*Solve

67log5

945 34/1 x

*Objectives

*Use the exponential function y=ex

*Find natural logarithms of numbers

*Solve equations and inequalities using natural logarithms

*Solve real-world applications with natural logarithms.

The formula for exponential growth and decay (in terms

of e) is : where A is the final amount, P is the

initial (principal) amount, r is the rate constant, and t is

time. (HINT: Look for the phrase “compounded

continuously” when using this formula.)

rtPeA

*On your calculator, find the e button. What number does the calculator give you?

2.718

The number e is JUST A NUMBER…just like ……….JUST A NUMBER.

*Compare the balance after 25 years of a $10,000 investment earning 6.75% interest compounded continuously to the same investment compounded semiannually.

)25)(0675(.10000ey )25)(2()2

0675.1(10000 y

48.54059 61.52574

According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume that the initial temperature is 90oF and that r=0.275.

Write a function to model the rate at which the liquid cools.

))(275(.90 te

b. Find the temperature of the liquid after 4 minutes

c. Graph the function and use the graph to verify your answer to part (b).

37.27090 )4)(275(. e

skip

*A natural logarithm is a logarithm with a base of e and is denoted ln x.

Convert to a natural logarithm and evaluate. 254log6

09.36ln

254ln

*Solve: 6.5 = -16.25 ln x

xln4. xe 4.

67.x

Solve each equation or inequality by using natural logarithms.

a. 32x = 7x-1 b. 486 22

x

12 7ln3ln xx

7ln)1(3ln2 xx

7ln7ln)3ln2( xx

7ln)7ln3ln2( x

7ln3ln2

7ln

x

74.7x

48ln6ln)2( 2 x

6ln

48ln22 x

16.42 x

016.42 x

04.204.2 x