3.5Exponential and Logarithmic Models

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IntroductionCommon types of mathematical models involving exponential functions or logarithmic functions include:

1. Exponential growth model: y = aebx, b > 0

Exponential decay model: y = aebx, b < 0

2. Logistic growth model:

3. Gaussian model: y = ae

4. Logarithmic models: y = a + blnx ; y = a + blog10x

5. Newton’s Law of Cooling:

6. Exponential Regression model:

0( ) , 0ktt m mT T T T e k

xy ab

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1. Exponential Growth/Decay Model

Growth: y = aebx

, b > 0 Decay: y = aebx

, b < 0

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0kt

fA A eAo = initial value

Af = final value

k = rate of increase if k > 0;

rate of decrease if k < 0

t = time between Ao and Af

Exponential Growth/Decay Model

The following is a more user-friendly model to use when problem solving:

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Example 1 – Demography

Estimates of the world population (in millions) from 2003 through 2009 are shown in the table. A scatter plot of the data is shown below. (Source: U.S. Census Bureau)

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Example 1 – Demography

An exponential growth model that approximates these data is given by

P = 6097e0.0116t, 3 t 9

where P is the population (in millions) and t = 3 represents 2003.

a) According to this model, when will the world population reach 7.1 billion?

cont’d

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Example 1 – Solution

P = 6097e0.0116t

7100 = 6097e0.0116t

1.16451 e0.0116t

In1.16451 Ine0.0116t

0.15230 0.0116t

t 13.1

According to the model, the world population will reach 7.1 billion in 2013.

cont’d

Write original equation.

Substitute 7100 for P.

Divide each side by 6097.

Take natural log of each side.

Inverse Property

Divide each side by 0.0116.

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2. Logistic Growth Model

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Logistic Growth Model

( )1 bt

cP t

aeP(t) = population after time t

c = carrying capacity

a = constant

b = growth rate

t = time

The following is a more user-friendly model to use when problem solving:

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Logistic Growth Model

Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph below.

Logistic Curve

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Logistic Growth Model

One model for describing this type of growth pattern is the logistic curve given by the function

where y is the population size and x is the time. An example is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve (or S curve).

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Example 2 – Spread of a Virus

On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by

where y is the total number of students infected after days. The college will cancel classes when 40% or more of the students are infected.

a. How many students are infected after 5 days?

b. After how many days will the college cancel classes?

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Example 2 – Solution

a. After 5 days, the number of students infected is

54.

b. Classes are canceled when the number of infected students is (0.40)(5000) = 2000.

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Example 2 – Solution

1 + 4999e –0.8t = 2.5

e –0.8t =

In e –0.8t = In

– 0.8t = In

t = 10.14

So, after about 10 days, at least 40% of the students will be infected, and classes will be canceled.

cont’d

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3. Gaussian Model

2( )

x bcy ae

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Gaussian Model

This type of model is commonly used in probability and statistics to represent populations that are normally distributed. For standard normal distributions, the model takes the form

The graph of a Gaussian model is called a bell-shaped curve.

Try graphing the normal distribution curve with a graphing utility.

Can you see why it is called a bell-shaped curve?

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Gaussian Model

The average value for a population can be found from the bell-shaped curve by observing where the maximum y-value of the function occurs. The x-value corresponding to the maximum y-value of the function represents the average value of the independent variable—in this case, x.

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Example 3 – SAT Scores

In 2009, the Scholastic Aptitude Test (SAT) mathematics scores for college-bound seniors roughly followed the normal distribution

where x is the SAT score for mathematics.

a) Use a graphing utility to graph this function

b) Estimate the average SAT score.

2( 515)

26,9120.0034 , 200 800x

y e x

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Example 3 – Solution

On this bell-shaped curve, the maximum value of the curve represents the average score. Using the maximum feature of the graphing utility, you can see that the average mathematics score for college bound seniors in 2009 was 515.

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4. Logarithmic Models

y = a + b ln x y = a + b log10x

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Example 4 - Meteorology

In meteorology, the relationship between the height H of a weather balloon (measured in km) and the atmospheric pressure p (measured in millimeters of mercury) is modeled by the function

a) Predict the height of a weather balloon when the atmospheric pressure is 560 millimeters of mercury.

b) If the height of the balloon is 3 km, what is the atmospheric pressure?

c) Graph this model. Does it look like a log graph? Explain.

pH ln848

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5. Newton’s Law of Cooling

0( ) , 0ktt m mT T T T e k

Tt = temperature of object at time t

Tm = temperature of surrounding medium (room temp)

To = initial temperature of heated object

k = negative constant

t = time

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Example 5 – Cooling Heated Object

An object is heated to 100°C and is then allowed to cool ina room whose air temperature is 30°C.

a) If the temp of the object is 80°C after 5 minutes, find the value of k.

b) Determine the time that needs to elapse before the object is 75°C.

c) Graph the relation found between temperature and time.

d) Using the graph, determine the time that needs to elapse before the object is 75°C.

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6. Exponential Regression Modelon TI-84

xy aba = initial value

b = ratio of successive y-values