3.5 EXPONENTIAL AND LOGARITHMIC MODELS

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• Recognize the five most common types of

models involving exponential and logarithmic

functions.

• Use exponential growth and decay functions to

model and solve real-life problems.

• Use Gaussian functions to model and solve

real-life problems.

• Use logistic growth functions to model and solve

real-life problems.

• Use logarithmic functions to model and solve

real-life problems.

What You Should Learn

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Introduction

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Introduction

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

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

3. Gaussian model: 𝒚 = 𝒂𝒆− 𝒙−𝒃 𝟐/𝒄

4. Logistic growth model: 𝒚 =𝒂

𝟏+𝒃𝒆−𝒓𝒙

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

𝒚 = 𝒂 + 𝒃 𝐥𝐨𝐠 𝒙

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Introduction

Exponential growth model Exponential decay model Gaussian model

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Introduction

Logistic growth model Natural logarithmic model Common logarithmic model

cont’d

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Exponential Growth and Decay

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Example 1 – Online Advertising

Estimates of the amounts (in billions of dollars) of U.S.

online advertising spending from 2007 through 2011 are

shown in the table.

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Example 1 – Online Advertising

A scatter plot of the data is shown In Figure 3.34.

(Source: eMarketer)

An exponential growth model that

approximates these data is given by

S = 10.33e0.1022t, 7 t 11, where S

is the amount of spending (in billions)

and t = 7 represents 2007.

Online Advertising Spending

Figure 3.34

cont’d

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Example 1 – Online Advertising

Compare the values given by the model with the estimates

shown in the table. According to this model, when will the

amount of U.S. online advertising spending reach $40 billion?

Solution:

cont’d

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Example 1 – Solution

To find when the amount of U.S. online advertising

spending will reach $40 billion, let S = 40 in the model and

solve for t.

10.33e0.1022t = S

10.33e0.1022t = 40

e0.1022t 3.8722

ln e0.1022t ln 3.8722

0.1022t 1.3538

Write original model.

Substitute 40 for S.

Divide each side by 10.33.

Take natural log of each side.

Inverse Property

cont’d

t 13.2 Divide each side by 0.1022.

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Gaussian Models

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Gaussian Models

The Gaussian models are of the form

𝑦 = 𝑎𝑒−𝑥−𝑏 2

𝑐

commonly used: probability and statistics to represent

populations that are normally distributed.

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

curve.

The standard normal distribution: 𝑦 =1

2𝜋𝑒−

𝑥2

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The average value of a population can be found from the

bell-shaped curve by observing the maximum y-value of

the function occurs.

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Example 4 – SAT Scores

In 2008, the Scholastic Aptitude Test (SAT) math scores for

college-bound seniors roughly followed the normal

distribution given by

y = 0.0034e –(x – 515)2/26,912, 200 x 800

where x is the SAT score for mathematics. Sketch the

graph of this function. From the graph, estimate the

average SAT score. (Source: College Board)

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Logistic Growth Models

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Logistic Growth Models

the logistic curve given by the function

where y is the population size and

x is the time.

A logistic growth curve is also called

a sigmoidal curve.

Figure 3.40

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Example 5 – Spread of a Virus

On a college campus of 5000 students, one student returns

from vacation with a contagious and long-lasting flu virus.

The spread of the virus is modeled by

, t 0

where y is the total number of students infected after t 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 5(a) – Solution

After 5 days, the number of students infected is

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Example 5(b) – Solution

Classes are canceled when the number infected is

(0.40)(5000) = 2000.

cont’d

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Example 5(b) – Solution

So, after about 10 days, at least 40% of the students will

be infected, and the college will cancel classes.

cont’d

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Logarithmic Models

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Example 6 – Magnitudes of Earthquakes

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.

Find the intensity of each earthquake. (Intensity is a

measure of the wave energy of an earthquake.)

a. Nevada in 2008: R = 6.0

b. Eastern Sichuan, China in 2008: R = 7.9

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Example 6 – Solution

a. Because I0 = 1 and R = 6.0, you have

6.0 =

106.0 = 10log I

I = 106.0

= 1,000,000.

Substitute 1 for I0 and 6.0 for R.

Exponentiate each side.

Inverse Property

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Example 6 – Solution

b. For R = 7.9, you have

7.9 =

107.9 = 10log I

I = 107.9

79,400,000.

cont’d

Substitute 1 for I0 and 7.9 for R.

Exponentiate each side.

Inverse Property

The intensity of the earthquake

in Eastern Sichuan was about

79 times as great as that of the

earthquake in Nevada.