copyright © 2006 brooks/cole, a division of thomson learning, inc. exponential and logarithmic...

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Exponential and Logarithmic Functions

5• Exponential Functions

• Logarithmic Functions

• Compound Interest

• Differentiation of Exponential Functions

• Differentiation of Logarithmic Functions

• Exponential Functions as Mathematical Models


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

x

y


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


Properties of the Exponential Function

,

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, 1xy f x b b b

,

,


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


Logarithms

log if and only if 0yby x x b x

The logarithm of x to the base b is defined by

Ex.

43

07

-2

1/ 3

15

log 81 4; 3 81

log 1 0; 7 1

1log 9 2; 81

3

log 5 1; 5 5


ExamplesEx. Solve each equation

a.

b.

2log 5x 52 32x

27log 3 x3 27x

33 3 x1 3x1

3x

m na a m n


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


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


Logarithmic Function

( ) log 0, 1bf x x b b

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

Properties:

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

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

Decreasing on (0, ) if b < 1

,


Graphs of Logarithmic Functions

Ex. 3( ) logf x x

(1,0)

3xy

3logy x

1

3

x

y

1/ 3logy x

1/ 3( ) logf x x

x x

yy

(1,0)

(0, 1) (0, 1)


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.


ExampleA normal child’s systolic blood pressure may be approximated by the function

where p(x) is measured in millimeters of mercury, x is measured in pounds, and m and b are constants. Given that m = 19.4 and b = 18, determine the systolic blood pressure of a child who weighs 92 lb.

( ) (ln )p x m x b

Since 19.4, 92, and 18

we have (92) 19.4(ln 92) 18

105.72

m x b

p


Compound Interest Formula

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


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


ExampleHow long will it take an investment of $10,000 to grow to $15,000 if it earns an interest at the rate of 12% / year compounded quarterly?

40.12

1 15,000 10,000 14

mt tr

A Pm

4 41.5 1.03 ln(1.03) ln1.5

4 ln1.0

Taking ln both sides

log l3 ln1.5

ln1.5 3.43 years

4ln1.0

og

3

nb

t t

bm n mt

t


Effective Rate of Interest

eff 1 1m

rr

m

reff = Effective rate of interest r = Nominal interest rate/year m = 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

It is about 6.7% per year.


Present Value Formula for Compound Interest

1mt

rP A

m

A = The accumulated amount after mt periods

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


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


Continuous Compound Interest Formula

rtA Pe

A = The accumulated amount after t years

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


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


Differentiation of Exponential Functions

x xde e

dx

( ) ( ) ( )f x f xde e f x

dx

Chain Rule for Exponential Functions

Derivative of Exponential Function

If f (x) is a differentiable function, then


ExamplesFind the derivative of

3 5( ) .xf x e

3 4 4 4( ) 4 4x xf x x e x e 3 44 1xx e x

Find the relative extrema of 4 4( ) .xf x x e

3 5( ) 3 5x df x e x

dx

3 55 xe

-1 0

+ – +

Relative Min. f (0) = 0Relative Max. f (-1) = 4

1

e

f

x


Differentiation of Logarithmic Functions

1ln 0

dx x

dx x

( )ln ( ) ( ) 0

( )

d f xf x f x

dx f x

Chain Rule for Exponential Functions

Derivative of Exponential Function

If f (x) is a differentiable function, then


ExamplesFind the derivative of 2( ) ln 2 1 .f x x

1( ) 2f x

x

Find an equation of the tangent line to the graph of ( ) 2 ln at 1, 2 .f x x x

2

2

2 1( )

2 1

dx

dxf xx

2

4

2 1

x

x

(1) 3f

2 3( 1)

3 1

y x

y x

Slope:

Equation:


Logarithmic Differentiation

1. Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms.

2. Differentiate both sides of the equation with respect to x.

3. Solve the resulting equation for .dy

dx


ExamplesUse logarithmic differentiation to find the derivative of 5

3 2 9 1y x x

5ln ln 3 2 9 1y x x

5ln ln 3 2 ln 9 1y x x

1ln ln 3 2 5ln 9 1

2y x x

1 3 5(9)

2 3 2 9 1

dy

y dx x x

5 3 45

3 2 9 12 3 2 9 1

dyx x

dx x x

Apply ln

Differentiate

Properties of ln

Solve


Exponential Growth/Decay Models

0( ) 0ktQ t Q e t

Q0 is the initial quantity

k is the growth/decay constant

A quantity Q whose rate of growth/decay at any time t is directly proportional to the amount present at time t can be modeled by:

Growth

Decay 0( ) 0ktQ t Q e t


ExampleA certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally and after 4 hours there were 120, find the number of bacteria present after 6 hours.

0 20 soQ ( ) 20 ktQ t e4120 20 ke

ln 60.4479

4k

0.4479(6)(6) 20 294Q e


Learning Curves

( ) 0ktQ t C Ae t

C, A, k are positive constants

An exponential function may be applied to certain types of learning processes with the model:

0,C A

y = C

x

y


ExampleSuppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 0.3( ) 200 150 tT t e

1. Find the temperature of the object after 5 minutes.0.3(5)(5) 200 150 166.5T e

2. Find the time it takes for the temperature of the object to reach 190°.

0.3190 200 150 te 0.31/15 te ln 1/15

9 min.0.3

t


Logistic Growth Model

A, B, k are positive constants

An exponential function may be applied to a logistic growth model:

( ) 01 kt

AQ t t

Be

y = A

0,1

A

B

x

y


Example

2400( )

1 1199 ktR t

e

The number of people R, in a small school district who have heard a particular rumor after t days can be modeled by:

If 10 people know the rumor after 1 day, find the number who heard it after 6 days.

(1)

240010

1 1199 ke

1 1199 240ke 239

ln 1.6131199

k …


1.613

2400( )

1 1199 tR t

e

So

1.613(6)

2400(6) 2232 people

1 1199R

e


Example

5000( )

1 1249 ktQ t

e

The number of soldiers at Fort MacArthur who contracted influenza after t days during a flu epidemic is approximated by the exponential model:

If 40 soldiers contracted the flu by day 7, find how many soldiers contracted the flu by day 15.

(7)

500040

1 1249 ke

71 1249 125ke 1 124

ln 0.337 1249

k …


0.33

5000( )

1 1249 tQ t

e

So

0.33(15)

5000(15) 508 people

1 1249

So approximately 508 soldiers contracted

the flu by day 15.

Qe

copyright © 2006 brooks/cole, a division of thomson learning, inc. exponential and logarithmic...

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