1. Copyright © Cengage Learning. All rights reserved. Logarithmic Function Modeling SECTION 6.5

2. 2 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

3. 3 Graphing Logarithmic Functions

4. 4 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.

5. 5 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

6. 6

7. 7 To further understand the logarithmic model we graph the equation below. Figure 6.14

8. 8 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

9. 9 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.

10. 10 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 = log b (x) with b > 1 concave down and increasing (b) y = log b (x) with 0 < b < 1 concave up and decreasing

11. 11 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

12. 12 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

13. 13 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

14. 14 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

15. 15 Finding an Exponential Model Using Logarithms

16. 16 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.

17. 17 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

18. 18 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

19. 19 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

20. 20 Finding an Exponential Model Using Logarithms The steps to find an exponential model by linearizing a data set are summarized below.

21. 21 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

22. 22 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

23. 23 Example 3(a) – Solution We create Table 6.28 to calculate log(N). Table 6.28

24. 24 Example 3(b) – Solution Using linear regression on the data in columns t and log(N), we obtain cont’d

25. 25 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