Modeling with Exponential and

Logarithmic Functions

The mathematical model for exponential growth or decay is given by

f (t) = A0ekt or A = A0ekt.• If k > 0, the function models the amount or size of a growing entity. A0

is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate.

• If k < 0, the function models the amount or size of a decaying entity. A0 is the original amount or size of the decaying entity at time t = 0. A is the amount at time t, and k is a constant representing the decay rate.

The mathematical model for exponential growth or decay is given by

f (t) = A0ekt or A = A0ekt.• If k > 0, the function models the amount or size of a growing entity. A0

is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate.

• If k < 0, the function models the amount or size of a decaying entity. A0 is the original amount or size of the decaying entity at time t = 0. A is the amount at time t, and k is a constant representing the decay rate.

decreasingA0

x

yincreasing

y = A0ekt

k > 0

x

y

y = A0ekt

k < 0A0

Exponential Growth and Decay Models

The graph below shows the growth of the Mexico City metropolitan area from 1970 through 2000. In 1970, the population of Mexico City was 9.4 million. By 1990, it had grown to 20.2 million.

• Find the exponential growth function that models the data.• By what year will the population reach 40 million?

2015

105

2530

1970 1980 1990 2000

Pop

ulat

ion

(mil

lion

s)

Year

Example

Solution

a. We use the exponential growth modelA = A0ekt

in which t is the number of years since 1970. This means that 1970 corresponds to t = 0. At that time there were 9.4 million inhabitants, so we substitute 9.4 for A0 in the growth model.

A = 9.4 ekt

We are given that there were 20.2 million inhabitants in 1990. Because 1990 is 20 years after 1970, when t = 20 the value of A is 20.2. Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k > 0 because the problem involves growth.

A = 9.4 ekt Use the growth model with A0 = 9.4.

20.2 = 9.4 ek•20 When t = 20, A = 20.2. Substitute these values.

Example cont.

Solution

We substitute 0.038 for k in the growth model to obtain the exponential growth function for Mexico City. It is A = 9.4 e0.038t where t is measured in years since 1970.

Example cont.20.2/ 9.4 = ek•20 Isolate the exponential factor by dividing both sides by 9.4.

ln(20.2/ 9.4) = lnek•20 Take the natural logarithm on both sides.

0.038 = k Divide both sides by 20 and solve for k.

20.2/ 9.4 = 20k Simplify the right side by using ln ex = x.

Solution

b. To find the year in which the population will grow to 40 million, we substitute 40 in for A in the model from part (a) and solve for t.

Because 38 is the number of years after 1970, the model indicates that the population of Mexico City will reach 40 million by 2008 (1970 + 38).

A = 9.4 e0.038t This is the model from part (a).

40 = 9.4 e0.038t Substitute 40 for A.

Example cont.

ln(40/9.4) = lne0.038t Take the natural logarithm on both sides.

ln(40/9.4)/0.038 =t Solve for t by dividing both sides by 0.038

ln(40/9.4) =0.038t Simplify the right side by using ln ex = x.

40/9.4 = e0.038t Divide both sides by 9.4.

• Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14.

• In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.

Solution

We begin with the exponential decay model A = A0ekt. We know that k < 0 because the problem involves the decay of carbon-14. After 5715 years (t = 5715), the amount of carbon-14 present, A, is half of the original amount A0. Thus we can substitute A0/2 for A in the exponential decay model. This will enable us to find k, the decay rate.

Text Example

Substituting for k in the decay model, the model for carbon-14 is A = A0e –0.000121t.

k = ln(1/2)/5715=-0.000121 Solve for k.

1/2= ek5715 Divide both sides of the equation by A0.

Solution A0/2= A0ek5715 After 5715 years, A = A0/2

ln(1/2) = ln ek5715 Take the natural logarithm on both sides.

ln(1/2) = 5715k ln ex = x.

Text Example cont.

Solution

The Dead Sea Scrolls are approximately 2268 years old plus the number of years between 1947 and the current year.

A = A0e-0.000121t This is the decay model for carbon-14.

0.76A0 = A0e-0.000121t A = .76A0 since 76% of the initial amount remains.

0.76 = e-0.000121t Divide both sides of the equation by A0.

ln 0.76 = ln e-0.000121t Take the natural logarithm on both sides.

ln 0.76 = -0.000121t ln ex = x.

Text Example cont.

t=ln(0.76)/(-0.000121) Solver for t.

Logistic Growth Model

• The mathematical model for limited logistic growth is given by

• Where a, b, and c are constants, with c > 0 and b > 0. c is the limiting size!

f (t) c

1 ae bt or A c

1 ae bt

tetf

5.1201

000,30)(

a)How many people became ill when the epidemic began?

This function describes the number of people who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants

)0(5.1201

000,30)0(

ef

201

000,30)0(

f

1429)0( f

This function describes the number of people who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants

b)How many people were ill be the end of the 4th week?

tetf

5.1201

000,30)(

)4(5.1201

000,30)4(

ef

583,28)4( f

This function describes the number of people who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants

tetf

5.1201

000,30)(

c)What is the limiting size of f(t)?

c is the limiting size of a logistic growth model; 30,000

Newton’s Law of Cooling

The temperature, T, of a heated object at time t is given by

T = C + (T0 - C)ekt

Where C is the constant temperature of the surrounding medium, T0 is the initial temperature of the heated object, and k is a negative constant that is associated with the cooling object.

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees.

a)Find a model for the temp, T, after t minutes.

b)What is the temp of the cake after 40 minutes?

c)When will the temp be 90 degrees?

kteCTCT )( 0 kteT )70210(70

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees.

a)Find a model for the temp, T, after t minutes.

30)140(70140 ke30)140(70 ke

30

2

1 ke

k302

1ln

k30

21

ln

k 0231.teT 0231.)140(70

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees.

b)What is the temp of the cake after 40 minutes?

teT 0231.)140(70

)40(0231.)140(70 eT

126T

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees.

c)When will the temp be 90 degrees?

teT 0231.)140(70 te 0231.)140(7090

te 0231.)140(20

te 0231.

7

1

t0231.7

1ln

t 0231.

71

ln

t84

When a murder is committed, the body, originally 37C, cools according to Newton’s Law of Cooling. Suppose on the day of a very difficult exam your math teacher’s body is found at 2pm with a body temperature of 32C in a room with a constant temperature of 20C. 3 hours later the body temperature is 27C. Being the bright student you are, find an equation to model this situation and determine at what time the murder was committed.

Expressing an Exponential Model in Base e.

• y = abx is equivalent to y = ae(lnb)x

xy )017.1(557.2 Rewrite

xey )017.1(ln557.2

xey 017.557.2

Example

• The value of houses in your neighborhood follows a pattern of exponential growth. In the year 2000, you purchased a house in this neighborhood. The value of your house, in thousands of dollars, t years after 2000 is given by the exponential growth model V = 125e.07t

• When will your house be worth $200,000?

ExampleSolution:V = 125e.07t

200 = 125e.07t

1.6 = e.07t

ln1.6 = ln e.07t

ln 1.6 = .07tln 1.6 / .07 = t6.71 = t