Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning,

Inc.

Exponential and Logarithmic Functions

5• Exponential Functions

• Logarithmic Functions

• Compound Interest

• Models


Inc.

Exponential Function

( ) 0, 1xf x b b b

An exponential function with base b and exponent x is defined by

Ex. ( ) 3xf x Domain: All reals

Range: y > 0(0,1)

( )y f x

0 1

1 3 2 9

11 3

x y


Inc.

Laws of ExponentsLaw Example

1. x y x yb b b

2.x

x yy

bb

b

4.x x xab a b

3.yx xyb b

5.x x

x

a a

b b

1/ 2 5 / 2 6 / 2 32 2 2 2 8 12

12 3 93

55 5

5

61/ 3 6 / 3 2 18 8 8

64

3 3 3 32 2 8m m m 1/ 3 1/ 3

1/ 3

8 8 2

27 327


Inc.

Properties of Exponential Functions

,

1. The domain is .

2. The range is (0, ).

3. It passes through (0, 1).

4. It is continuous everywhere.

5. If b > 1 it is increasing on . If b < 1 it is decreasing on .

( ) 0, 1xf x b b b

,

,


Inc.

ExamplesEx. Simplify the expression.

Ex. Solve the equation

42 1/ 2

3 7

3x y

x y

4 8 2

3 7

3 x y

x y

5

5

81x

y

3 1 4 24 2x x 2 3 1 4 22 2x x

6 2 4 22 2x x 6 2 4 2x x

2 4x 2x


Inc.

Logarithms

log if and only if 0yby x x b x

An logarithmic of x to the base b is defined by

Ex. 3

7

1/ 3

5

log 81 4

log 1 0

log 9 2

log 5 1


Inc.

ExamplesEx. Solve each equation

a.

b.

2log 5x 52 32x

27log 3 x3 27x

33 3 x1 3x1

3x

m na a m n


Inc.

Notation: Common LogarithmNatural Logarithm

10log log

ln loge

x x

x x

Laws of Logarithms1. log log log

2. log log log

3. log log

4. log 1 0

5. log 1

b b b

b b b

nb b

b

b

mn m n

mm n

n

m n m

b


Inc.

ExampleUse the laws of logarithms to simplify the expression:

7 1/ 25 5 5 5log 25 log log logx y z

7

525

logx y

z

5 5 51

2 7 log log log2

x y z


Inc.

Logarithmic Function

( ) log 0, 1bf x x b b

An logarithmic function of x to the base b is defined by

Properties

1. Domain: (0, )2. Range:

3. Intercept: (1, 0)4. Continuous on (0, )5. Increasing on (0, ) if b > 1

Decreasing on (0, ) if b < 1

,


Inc.

Logarithmic Function Graphs

Ex. 3( ) logf x x

(1,0)

3xy

3logy x

1

3

x

y

1/ 3logy x

1/ 3( ) logf x x


Inc.

ln 0

ln for any real number

x

x

e x x

e x x

and lnxe x

Ex. Solve 2 1110

3xe

2 1 30xe 2 1 ln(30)x

ln(30) 11.2

2x

Apply ln to both sides.


Inc.

Compound Interest

1mt

rA P

m

A = the accumulated amount after mt periods

P = Principalr = Nominal interest rate per yearm = Number of periods/yeart = Number of years


Inc.

ExampleFind the accumulated amount of money after 5 years if $4300 is invested at 6% per year compounded quarterly.

1mt

rA P

m

4(5).06

4300 14

A

= $5791.48


Inc.

Effective Rate of Interest

eff 1 1m

rr

m

reff = Effective rate of interestr = Nominal interest rate/yearm = number of conversion periods/year

Ex. Find the effective rate of interest corresponding to a nominal rate of 6.5% per year, compounded monthly.

eff 1 1m

rr

m

12.065

1 112

.06697

So about 6.67% per year.


Inc.

Present Value for Compound Interest

1mt

rP A

m

A = the accumulated amount after mt periods

P = Principalr = Nominal interest rate per yearm = Number of periods/yeart = Number of years


Inc.

ExampleFind the present value of $4800 due in 6 years at an interest rate of 9% per year compounded monthly.

1mt

rP A

m

12(6).09

4800 112

P

$2802.83


Inc.

Continuous Compound Interest

P rtA e

A = the accumulated amount after t years

P = Principalr = Nominal interest rate per yeart = Number of years


Inc.

ExampleFind the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.

rtA Pe0.12(25)7500A e

$150,641.53

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions.

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