In Lesson 7A, you considered the characteristics of logarithmic functions and their corresponding graphs. This lesson introduces logarithmic functions as mathematical models. Beginning with a realistic situation, you will use a logarithmic function to construct a regression model and use the model to answer questions about the situation.

The next example reviews how to graph a scatter plot and corresponding regression model using the graphing calculator. As you read the example, complete the steps on your calculator. Contact your teacher if you have any difficulty following the examples with your calculator.

In February 2004, Mark Zuckerberg launched Facebook from his Harvard dorm room. Facebook had 1 000 000 registered users by December of that year, and it has been growing rapidly ever since, as shown in the table below. Use the data to determine the equation of the logarithmic regression function for the time, t, in months, as a function of the number of registered users, n.
Number of Months since Feb., 2004
Number of Registered Users (millions), n
Numbers of Months since Feb., 2004
Number of Registered Users (millions), n
10
1.0
62
200.0
22
5.5
65
250.0
34
12.0
67
300.0
38
20.0
70
350.0
44
50.0
72
450.0
54
100.0
77
500.0
59
150.0
79
550.0
60
175.0

Note: To find a regression equation only, omit steps 1, 2, 3, and 7.

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Step 2: Clear the functions; press Y= then, CLEAR for each function that must be deleted.

Step 3: Select values for the viewing window; press WINDOW. Use the keypad to type the values from the table. Use the up and down arrow keys to scroll through the list.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted; then, CLEAR, ENTER. Repeat Step 5 for all columns that have unneeded data. Use the left and right arrow keys to move between columns.

Step 6: Enter the data.

Step 7: View the data; press GRAPH.

Step 8: Choose the regression model; press STAT, right arrow.

Step 9: Select the logarithmic regression; press 9.

Step 10: Place the logarithmic regression into the function area; press VARS, right arrow, ENTER, ENTER, ENTER.

Step 11: View the scatter plot and regression model; press GRAPH.

The logarithmic regression equation is y = 5.888 + 10.8333 ln x.

As mentioned in Warm Up, Example 1 shows the logarithmic regression model is a natural logarithm. Because this is the logarithm found most often in math and science applications, the natural logarithm is the standard used for the regression model in most calculators and computer programs.

The form of the natural logarithm used for regression models is y = a + b ln x.

Now that you are familiar with the logarithmic regression, use it to solve problems. Example 2 explores this.