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3 Exponential and Logarithmic Functions

Copyright © Cengage Learning. All rights reserved.

3.5 Exponential and Logarithmic Models

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What You Should Learn

• Recognize the five most common types of models involving exponential or logarithmic functions.

• Use exponential growth and decay functions to model and solve real-life problems.

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What You Should Learn

• Use logistic growth functions to model and solve real-life problems.

• Use logarithmic functions to model and solve real-life problems.

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Introduction

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Introduction

There are many examples of exponential and logarithmic models in real life.

The five most common types of mathematical models involving exponential functions or logarithmic functions are as follows.

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

2. Exponential decay model: y = ae–bx, b > 0

7

Introduction

3. Gaussian model: y = ae

4. Logistic growth model:

5. Logarithmic models: y = a + b ln x,

y = a + b log10x

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Introduction

The basic shapes of these graphs are shown in Figure 3.32.

Figure 3.32

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Introduction

Figure 3.32

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Exponential Growth and Decay

Please read the next three slides, but do not copy them down.

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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 in Figure 3.33. (Source: U.S. Census Bureau)

Figure 3.33

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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. Compare the values given by the model with the estimates shown in the table. According to this model, when will the world population reach 7.1 billion?

cont’d

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Example 1 – Solution

The following table compares the two sets of population figures.

From the table, it appears that the model is a good fit for the data. To find when the world population will reach 7.1 billion, let

P = 7100

in the model and solve for t.

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Example 1 – Solution

6097e0.0116t = P

6097e0.0116t = 7100

e0.0116t 1.16451

Ine0.0116t In1.16451

0.0116t 0.15230

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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Gaussian Models

Please read the next three slides, but do not copy them down.

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Gaussian Models

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

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 4 – SAT Scores

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

y = 0.0034e–(x – 515)226,912, 200 x 800

where x is the SAT score for mathematics. Use a graphing utility to graph this function and estimate the average SAT score. (Source: College Board)

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Example 4 – Solution

The graph of the function is shown in Figure 3.37. 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.

Figure 3.37

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Logistic Growth Models

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Logistic Growth Models

Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph in Figure 3.39.

Figure 3.39

Logistic Curve

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Logarithmic Models

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Logarithmic Models

On the Richter scale, the magnitude R of an earthquake of intensity I is given by

where I0 = 1 is the minimum intensity used for comparison. Intensity is a measure of the wave energy of an earthquake.

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Example 6 – Magnitudes of Earthquakes

In 2009, Crete, Greece experienced an earthquake that measured 6.4 on the Richter scale. Also in 2009, the north coast of Indonesia experienced an earthquake that measured 7.6 on the Richter scale. Find the intensity of each earthquake.

Solution:

Because I0 = 1 and R = 6.4, you have

106.4 = 10 log10

I 106.4 = I

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Example 6 – Solution

For R = 7.6 you have

107.6 = 10 log10

I 107.6 = I

cont’d