Chapter 3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter 3

Exponential, Logistic, and Logarithmic Functions


3.1Exponential and Logistic Functions


Quick Review

3

3

4 / 3

2-3

5

Evaluate the expression without using a calculator.

1. -125

272.64

3. 27Rewrite the expression using a single positive exponent.

4.

Use a calculator to evaluate the expression.

5. 3.71293

a


Quick Review Solutions

6

3

3

4 / 3

2-3


1. -125

272. 64

3. 27 Rewrite the expression using a single positive e

-5

3481

1xponent.

4.

Use a calculator to evaa

a

5

luate the expression.

5. 3.71293 1.3


What you’ll learn about

Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models

… and whyExponential and logistic functions model many growth patterns, including the growth of human and animal populations.


Exponential Functions

Let and be real number constants. An in is a function that can be written in the form ( ) , where is nonzero,

is positive, and 1. The constant is the

x

a b xf x a b a

b b a initial v

exponential function

of (the valueat 0), and is the .

alue fx b base


Example Finding an Exponential Function from its Table of Values

Determine formulas for the exponential function and whose values are given in the table below.

g h


Example Finding an Exponential Function from its Table of Values

Determine formulas for the exponential function and whose values are given in the table below.

g h

1

Because is exponential, ( ) . Because (0) 4, 4. Because (1) 4 12, the base 3. So, ( ) 4 3 .

x

x

g g x a b g ag b b g x

1

Because is exponential, ( ) . Because (0) 8, 8.

1Because (1) 8 2, the base 1/ 4. So, ( ) 8 .4

x

x

h h x a b h a

h b b h x


Exponential Growth and Decay

For any exponential function ( ) and any real number ,( 1) ( ).

If 0 and 1, the function is increasing and is an . The base is its .

If 0 an

xf x a b xf x b f x

a b fb

a

exponentialgrowth function growth factor

d 1, the function is decreasing and is an . The base is its .

b fb

exponentialdecay function decay factor


Example Transforming Exponential Functions

-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x




-2The graph of ( ) 2 is obtained by translating the graph of ( ) 2 by2 units to the right.

x xg x f x






-Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x

The graph of ( ) 2 is obtained by reflecting the graph of ( ) 2 acrossthe -axis.

x xg x f xy


The Natural Base e

1lim 1

x

xe

x


Exponential Functions and the Base e

Any exponential function ( ) can be rewritten as ( ) , for any appropriately chosen real number constant .If 0 and 0, ( ) is an exponential growth function.If 0 and 0, (

x kx

kx

f x a b f x a ek

a k f x a ea k f

) is an exponential decay function.kxx a e


Exponential Functions and the Base e



3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e



3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e

3The graph of ( ) is obtained by horizontally shrinking the graph of ( ) by a factor of 3.

x

x

g x ef x e


Logistic Growth Functions

Let , , , and be positive constants, with 1. A

in is a function that can be written in the form ( ) or 1

( ) where the constant is the 1

x

kx

a b c k bcx f xa b

cf x ca e

logistic growth function

limit to gr

owth.


3.2Exponential and Logistic Modeling


Quick Review

2

Convert the percent to decimal form or the decimal into a percent.1. 16%2. 0.053. Show how to increase 25 by 8% using a single multiplication.Solve the equation algebraically.4. 20 720Solve the equ

b

3

ation numerically.5. 123 7.872b



Convert the percent to decimal form or the decimal into a percent.1. 16% 2. 0.05 3. Show how to increase 25 by 8% using a single multiplication.Solve the equation algebraically.

0.165%

25 1 4

.082

3

. 20 720 Solve the equation numerically.5. 123 7.872

6

0. 4

b

b



Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models

… and whyExponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.


Constant Percentage Rate

Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

0

0 0 0

Time in years Population0 (0) initial population1 (1) (1 )2 (2) (1)

P PP P Pr P rP P

2

0

3

0

0

(1 ) (1 )3 (3) (2) (1 ) (1 ) ( ) (1 ) t

r P rP P r P r

t P t P r


Exponential Population Model

0 0

If a population is changing at a constant percentage rate each year, then( ) (1 ) , where is the initial population, is expressed as a decimal,

and is time in years.

t

P rP t P r P r

t


Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponentialgrowth function or exponential decay function, and find the constant percentrate of growth.

tP t


Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponentialgrowth function or exponential decay function, and find the constant percentrate of growth.

tP t

Because 1 1.021, .021 0. So, is an exponential growth functionwith a growth rate of 2.1%.

r r P


Example Finding an Exponential Function

Determine the exponential function with initial value=10, increasing at a rate of 5% per year.


Example Finding an Exponential Function

Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

0Because 10 and 5% 0.05, the function is ( ) 10(1 0.05) or

( ) 10(1.05) .

t

t

P r P tP t


Example Modeling Bacteria Growth

Suppose a culture of 200 bacteria is put into a petri dish and the culturedoubles every hour. Predict when the number of bacteria will be 350,000.


Example Modeling Bacteria Growth

Suppose a culture of 200 bacteria is put into a petri dish and the culturedoubles every hour. Predict when the number of bacteria will be 350,000.

2

400 200 2800 200 2

( ) 200 2 represents the bacteria population hr after it is placedin the petri dish. To find out when the population will reach 350,000, solve350,000 200 2 for using

t

t

P t t

t

a calculator.10.77 or about 10 hours and 46 minutes.t


Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.


Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

103

Let ( ) be the population (in millions) of the U.S. years after 1900.Using exponential regression, find a model ( ) 80.5514 1.01289 .To find the population in 2003 find (103) 80.5514 1.01289 3

t

P t tP t

P

01.3.


Maximum Sustainable Population

Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.


Example Modeling a Rumor

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

0 is the day the

rumor begins to spread.(a) How many students have heard the rumor by the end of Day 0?(b) How long does it take for 1000 students to hear the rumor?

t


Example Modeling a Rumor

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

0 is the day the

rumor begins to spread.(a) How many students have heard the rumor by the end of Day 0?(b) How long does it take for 1000 students to hear the rumor?

t

-0.9 ( 0 )(a) (0) 1500 /(1 29 ) 1500 /(1 29 1) 1500 / 30 50. So 50 students have heard the rumor by the end of day 0.

S e

-0.9(b) Solve 1000 1500 /(1 29 ) for .4.5. So 1000 students have heard the rumor half way

through the fifth day.

te tt


3.3Logarithmic Functions and Their

Graphs


Quick Review

-2

11

32

0

3

4

Evaluate the expression without using a calculator.1. 6

82. 2

3. 7Rewrite as a base raised to a rational number exponent.

14.

5. 10e



3 / 2

1/

-2

11

3

4

2

0

3

4


1. 6

82. 2

3. 7 Rewrite as a base raised to a rational number exponent.

14.

5. 10

136

2

1

10

ee



Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels

… and whyLogarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.


Changing Between Logarithmic and Exponential Form

If 0 and 0 1, then log ( ) if and only if .y

bx b y x b x


Inverses of Exponential Functions


Basic Properties of Logarithms

0

1

log

For 0 1, 0, and any real number . log 1 0 because 1. log 1 because . log because . because log log .b

b

b

y y y

b

x

b b

b x yb

b b bb y b b

b x x x


An Exponential Function and Its Inverse


Common Logarithm – Base 10

Logarithms with base 10 are called common logarithms.

The common logarithm log10x = log x. The common logarithm is the inverse of the

exponential function y = 10x.


Basic Properties of Common Logarithms

0

1

log

Let and be real numbers with 0. log1 0 because 10 1. log10 1 because 10 10. log10 because 10 10 . 10 because log log .

y y y

x

x y x

yx x x


Example Solving Simple Logarithmic Equations

Solve the equation by changing it to exponential form.log 4x


Example Solving Simple Logarithmic Equations

Solve the equation by changing it to exponential form.log 4x

410 10,000x


Basic Properties of Natural Logarithms

0

1

ln

Let and be real numbers with 0. ln1 0 because 1. ln 1 because . ln because . because ln ln .

y y y

x

x y xe

e e ee y e e

e x x x


Graphs of the Common and Natural Logarithm


Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of ( ) ln(2 ).

y xh x x


Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of ( ) ln(2 ).

y xh x x

( ) ln(2 ) ln[ ( 2)]. So obtain the graph of ( ) ln(2 - ) fromln by applying, in order, a reflection across the -axis followed by

a translation 2 units to the right.

h x x x h x xy x y


Decibels

0

2 12 2

0

The level of sound intensity in (dB) is

10log , where (beta) is the number of decibels,

is the sound intensity in W/m , and 10 W/m is thethreshold of human hearing (the qu

II

I I

decibels

ietest audible soundintensity).


3.4Properties of Logarithmic Functions


Quick Review

3

3

-2

3 3

2 2

1/ 22 4

3

Evaluate the expression without using a calculator.1. log102. ln 3. log 10Simplify the expression.

4.

5. 2

e

x yx y

x yx



3

3

-2

3 3

2 2

1/ 2

5

4 22 4

3

5

Evaluate the expression without using a calculator.1. log10 2. ln 3. log

3 3

10 -Simplify the expression.

4.

2

2

5. 2

e

x yx y

xy

xx y yx



Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data

… and whyThe applications of logarithms are based on their many special properties, so learn them well.


Properties of Logarithms

Let , , and be positve real numbers with 1, and any real number. : log ( ) log log

: log log log

: log ( ) log

b b b

b b b

c

b b

b R S b cRS R S

R R SS

R c R

Product rule

Quotient rule

Power rule


Example Proving the Product Rule for Logarithms

Prove log ( ) log log .

b b bRS R S


Example Proving the Product Rule for Logarithms

Prove log ( ) log log .

b b bRS R S

Let log and log . The corresponding exponential statementsare and . Therefore,

log ( ) change to logarithmic form log ( ) log log

b b

x y

x y

x y

b

b b b

x R y Sb R b S

RS b bRS b

RS x yRS R S


Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to writelog 3 as a sum of logarithms or multiple logarithms.

xx


Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to writelog 3 as a sum of logarithms or multiple logarithms.

xx

5 5log 3 log3 log

log3 5log

x x

x


Example Condensing a Logarithmic Expression

Assuming is positive, write 3ln ln 2 as a single logarithm.x x


Example Condensing a Logarithmic Expression

Assuming is positive, write 3ln ln 2 as a single logarithm.x x

3

3

3ln ln 2 ln ln 2

ln2

x xx


Change-of-Base Formula for Logarithms

For positive real numbers , , and with 1 and 1,log

log .log

a

b

a

a b x a bx

xb


Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.


Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.

3

log10 1log 10 2.096log3 log3


3.5Equation Solving and Modeling


Quick Review

3 1/ 3

2 / 2

Prove that each function in the given pair is the inverse of the other.1. ( ) and ( ) ln

2. ( ) log and ( ) 10Write the number in scientific notation.3. 123,400,000Write the number in

x

x

f x e g x x

f x x g x

8

-4

decimal form.4. 5.67 105. 8.91 10



1 / 33ln ln

2/ 2

3 1/ 3

2 / 2

Prove that each function in the given pair is the inverse of the other.

1. ( ) and ( ) ln

2. ( ) log and ( ) 10

( ( ))

( ( )) log 1

Write the numbe

0 log1

r

0

x x

x x

x

x

f x e g x x

f x x

f g x e e x

f g x xg x

8

-

8

4

in scientific notation.3. 123,400,000 Write the number in decimal form.4. 5.67 10 5. 8.9

1.234 10

1 10567,000,000

0.0 8 91 00



Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression

… and whyThe Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.


One-to-One Properties

For any exponential function ( ) , If , then .

For any logarithmic function ( ) log , If log log , then .

x

u v

b

b b

f x bb b u v

f x xu v u v


Example Solving an Exponential Equation Algebraically

/ 2

Solve 40 1/ 2 5.x


Example Solving an Exponential Equation Algebraically

/ 2

Solve 40 1/ 2 5.x

/ 2

/ 2

/ 2 3 3

40 1/ 2 5

11/ 2 divide by 408

1 1 1 1 2 2 8 2/ 2 3 one-to-one property

6

x

x

x

xx


Example Solving a Logarithmic Equation

3Solve log 3.x


Example Solving a Logarithmic Equation

3Solve log 3.x

3

3 3

3 3

log 3log log10

1010

xx

xx


Orders of Magnitude

The common logarithm of a positive quantity is its order of magnitude.

Orders of magnitude can be used to compare any like quantities: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude

bigger than Earmuff Junction with a population of 8.


Richter Scale

The Richter scale magnitude of an earthquake is

log , where is the amplitude in micrometers ( m)

of the vertical ground motion at the receiving station, is the period of the associated seis

RaR B aT

T

mic wave in seconds, and accounts for the weakening of the seismic wave with increasingdistance from the epicenter of the earthquake.

B


pH

In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H+]More acidic solutions have higher hydrogen-ion concentrations and lower pH values.


Newton’s Law of Cooling

0

An object that has been heated will cool to the temperature of the medium in which it is placed. The temperature of the object at time can be modeled by

( ) ( ) for an appropriate vakt

m m

T tT t T T T e

0

lue of , where the temperature of the surrounding medium, the temperature of the object.

This model assumes that the surrounding medium maintains a constanttemperature.

m

kTT


Example Newton’s Law of Cooling

A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC?


Example Newton’s Law of Cooling

A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC?

0

0

5

5

5

Given 100, 15, and (5) 55.( ) ( )

55 15 8540 85

4085

40ln 5850.1507...

m

kt

m m

k

k

k

T T TT t T T T e

ee

e

k

k

0.1507

0.1507

Now find when ( ) 25.25 15 8510 85

10ln 0.15078514.2min .

t

t

t T te

e

t

t


Regression Models Related by Logarithmic Re-Expression

Linear regression: y = ax + b Natural logarithmic regression: y = a + blnx Exponential regression: y = a·bx

Power regression: y = a·xb


Three Types of Logarithmic Re-Expression


Three Types of Logarithmic Re-Expression (cont’d)


Three Types of Logarithmic Re-Expression(cont’d)


3.6Mathematics of Finance


Quick Review

1. Find 3.4% of 70.2. What is one-third of 6.25%?3. 30 is what percent of 150?4. 28 is 35% of what number?5. How much does Allyson have at the end of 1 yearif she invests $400 at 3% simple interest?



1. Find 3.4% of 70.

2. What is one-third of 6.25%? 3. 30 is what percent of 150? 4. 28 is 35% of what number? 5. How much does Allyson have at the end of 1

2.38

0.0208320%8

yearif

0

she invests $400 at 3% simple interest? $412



Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value

… and whyThe mathematics of finance is the science of letting your money work for you – valuable information indeed!


Interest Compounded Annually

If a principal is invested at a fixed annual interest rate , calculated at the end of each year, then the value of the investment after years is

(1 ) , where is expressed as a decimal.n

P rn

A P r r


Interest Compounded k Times per Year

Suppose a principal is invested at an annual rate compounded times a year for years. Then / is the interest rate per compounding

period, and is the number of compounding periods. The amou

P rk t r k

kt nt

in the account after years is 1 .kt

A

rt A Pk


Example Compounding Monthly

Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.


Example Compounding Monthly

Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

12 ( 5 )

Let 400, 0.08, 12, and 5,

1

0.08 400 112

595.9382...So the value of Paul's investment after 5 years is $595.94.

kt

P r k t

rA Pk


Compound Interest – Value of an Investment

Suppose a principal is invested at a fixed annual interest rate . The valueof the investment after years is

1 when interest compounds k times per year,

when interest co

kt

rt

P rt

rA Pk

A Pe

mpounds continuously.


Example Compounding Continuously

Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.


Example Compounding Continuously

Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

0.08 ( 5 )

400, 0.08, and 5,

400 596.7298...So Paul's investment is worth $596.73.

rt

P r tA Pe

e


Annual Percentage Yield

A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.


Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?


Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

4

4

4

Let the equivalent APY. The value after one year is 3000(1 ).

0.04653000(1 ) 3000 14

0.0465(1 ) 14

0.04651 1 0.047317...4

The annual percentage yield is 4.73%.

x A x

x

x

x


Future Value of an Annuity

The future value of an annuity consisting of equal periodic paymentsof dollars at an interest rate per compounding period (payment interval) is

1 1.

n

FV nR i

iFV R

i


Present Value of an Annuity

The present value of an annuity consisting of equal paymentsof dollars at an interest rate per period (payment interval) is

1 1.

n

PV nR i

iPV R

i


Chapter Test

4-1. State whether ( ) 2 is an exponential growth function or an

exponential decay function, and describe its end behavior using limits.2. Find the exponential function that satisfies the conditio

xf x e

ns:Initial height = 18 cm, doubling every 3 weeks.3. Find the logistic function that satisfies the conditions:Initial value = 12, limit to growth = 30, passing through (2,20).4. Describe how to transf

2

2

orm the graph of log into the graph of( ) log ( 1) 2.

5. Solve for : 1.05 3.x

y xh x x

x


Chapter Test

6. Solve for : ln(3 4) - ln(2 1) 57. Find the amount accumulated after investing a principal for years at an interest rate compounded continuously.

8. The population of Preston is 89,000 and i

x x xA P

t r

s decreasing by 1.8% each year.(a) Write a function that models the population as a function of time .(b) Predict when the population will be 50,000?9. The half-life of a certain substance is 1.5 sec

t

0

. The initial amount of substance is grams. (a) Express the amount of substance remaining asa function of time .(b) How much of the substance is left after 1.5 sec?(c) How much of the substance

S

t

0

is left after 3 sec?(d) Determine if there was 1 g left after 1 min.

S


Chapter Test

10. If Joenita invests $1500 into a retirement account with an 8% interest ratecompounded quarterly, how long will it take this single payment to grow to$3750?


Chapter Test Solutions

-

4-1. State whether ( ) 2 is an exponential growth function or an exponential decay function, and exponential decay

describe its end behavior u; li

sing limits.

2. Find

m ( ) , lim

the e

(

x

) 2x x

x

f

f x e

x f x

/ 21

ponential function that satisfies the conditions:Initial height = 18 cm, doubling every 3 weeks. 3. Find the logistic function that satisfies the conditions:Initial value = 12, limit t

( 2

o

18

) xf x

2

2

0.55

growth = 30, passing through (2,20).

4. Describe how to transform the g( ) 30 /(1 1.5 )

translate right 1 unit, relect across the -araph of log into the graph of

( ) log ( 1) 2. xis,transl

x

y xh

f x e

xx x

5. Solve for : 1.ate up 2 un

05 3. its.

22.5171xx x



6. Solve for : ln(3 4) - ln(2 1) 5 7. Find the amount accumulated after investing a principal for years at an interest rate compounded continuously.

8. The population of

-0.49

Pres

15

t

rt

x x xA P

t r

x

Pe

on is 89,000 and is decreasing by 1.8% each year.(a) Write a function that models the population as a function of time .

(b) Predict when the population will be ( ) 89,000(0.9

50,000? 82)

31.74 year

tPt

t

0

/1.5

0

9. The half-life of a certain substance is 1.5 sec. The initial amount of substance is grams. (a) Express the amount of substance remaining as

a functi

s

1( )2

on of time .

(b) How

t

t t S

S

S

0

0

0

much of the substance is left after 1.5 sec? (c) How much of the substance is left after 3 sec? (d) Determine if there was 1 g left after 1 mi

/2/ 4

1,009,500 metric n. t ns

o

SS

S



10. If Joenita invests $1500 into a retirement account with an 8% interest ratecompounded quarterly, how long will it take this single payment to grow t

11.5o

$375 7 y0? ears

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