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Logarithmic Function Modeling

SECTION 6.5

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Learning Objectives

1 Graph logarithmic functions from equations and tables

2 Use logarithmic regression to model real-world data sets

3 Use logarithms to linearize exponential data to find an exponential model

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Graphing Logarithmic Functions

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Graphing Logarithmic Functions

We model United States government spending with the exponential function

where s represents the spending (in billion dollars) and t represents years since 1990. Using the model, we can determine the amount of government spending in a particular year.

By solving the equation for t, we can create a model that will give us the year in which a particular level of spending is projected to occur.

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The function models the number of years since 1990, t, in which government spending will be s billion dollars. This logarithmic function is the inverse of the exponential function

Graphing Logarithmic Functions

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Graphing Logarithmic Functions

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Graphing Logarithmic Functions

To further understand the logarithmic model we graph the equation below.

Figure 6.14

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Graphing Logarithmic Functions

The graph is increasing and concave down with a horizontal intercept at the initial 1990 level of government spending ($1872.6 billion).

Figure 6.14

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Graphing Logarithmic Functions

By learning the basic shapes of logarithmic function graphs, we can quickly determine from a scatter plot if a logarithmic model is appropriate for a particular real-world situation.

The shape of a logarithmic function graph depends on the base of the logarithm.

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Graphing Logarithmic Functions

However, regardless of the base, the graph will have a vertical asymptote at the vertical axis, as shown in Figure 6.15.

Figure 6.15

(a) y = logb(x) with b > 1 concave down and increasing

(b) y = logb(x) with 0 < b < 1 concave up and decreasing

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Example 1 – Using Regression to Find a Logarithmic Model for a Data Set

The data set in Table 6.23 and scatter plot in Figure 6.16 show the inflation rates of the top 10 countries with the lowest rates of inflation.

Table 6.23 Figure 6.16

Countries with Lowest Inflation Rates

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Example 1 – Using Regression to Find a Logarithmic Model for a Data Set

Determine if a logarithmic function model is appropriate for this situation. If a logarithmic function is appropriate, use regression to find the logarithmic model.

cont’d

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Example 1 – Solution

The data set and scatter plot appear to be more or less increasing and concave down.

Since the countries are listed in rank order, we know as the rank number increases the inflation rate will also increase (or remain the same).

A logarithmic model is appropriate for this situation.

Using the graphing calculator, we determine the logarithmic equation of best fit is

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Example 1 – Solution

A graph of the model and the data is shown in Figure 6.17.

cont’d

Figure 6.17

Countries with Lowest Inflation Rates

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Finding an Exponential Model Using Logarithms

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Finding an Exponential Model Using Logarithms

An exponential data set is characterized by a constant ratio for equally spaced values. Another way to detect if a data set is exponential is to take the logarithm of the output values, as shown in next Example.

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Example 2 – Using Logarithms to Linearize Data

Complete Table 6.24 by calculating the logarithm of each of the output values. Then identify the mathematical relationship between the resultant values of

Table 6.24

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Example 2 – Solution

We complete the table as shown in Table 6.25 and then look for a pattern by calculating the average rates of change.

Table 6.25

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Example 2 – Solution

Since has a constant rate of change, it must be a linear function.

We readily recognize that y is an increasing linear function with slope 0.3010 and initial value 0.4771.

cont’d

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Finding an Exponential Model Using Logarithms

The steps to find an exponential model by linearizing a data set are summarized below.

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Example 3 – Finding an Exponential Model from a Linearized Data Set

Table 6.27 shows the number of insecticide treated nets (ITNs) sold or distributed in the African region in the fight against malaria.

Table 6.27

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Example 3 – Finding an Exponential Model from a Linearized Data Set

a. Calculate log(N) at each data point.

b. Use regression to find the linear equation that relates

t and log(N).

c. Use the result from part (b) to find the exponential

equation that relates t and N.

cont’d

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Example 3(a) – Solution

We create Table 6.28 to calculate log(N).

Table 6.28

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Example 3(b) – Solution

Using linear regression on the data in columns t and log(N), we obtain

cont’d

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Example 3(c) – Solution

Rewriting in exponential form yields

So the exponential function model is thousand ITNs, where t is the number of years since 1999.

cont’d