Section 5.4

Exponential and

Logarithmic Models

Objectives

Model data with exponential functionsUse constant percent change to determine if data fit an exponential model

Compare quadratic and exponential models of dataModel data with logarithmic functions

Example

The monthly premiums for $250,000 in term-life insurance over a 10-year term period increase with age of the men purchasing the insurance. The monthly premium for nonsmoking males are shown.

Example (cont)

a. Graph the data in the table with x as age and y in dollars.

Solution

Example (cont)

b. Create an exponential function that models these premiums as a function of age.

Solution

Using technology gives the exponential model for the monthly premium, rounded to four decimal places, as

y = 4.0389(1.0946x) dollars

Example (cont)

c. Graph the data and the exponential function that models the data on the same axes.

SolutionThe graph shows the scatter plot of the data of the (rounded) exponential equation used to model it. The model appears to be a good, but not perfect, fit for the data.

Constant Percent ChangesIf the percent change of the outputs of a set of data is constant for equally spaced inputs, an exponential function will be a perfect fit for the data.

If the percent change of the outputs is approximately constant for equally spaced inputs, an exponential function will be an approximate fit for the data.

Example

Suppose a company develops a product that is released with great expectations and extensive advertising, but sales suffer because of bad word of mouth from dissatisfied customers.a. Use the monthly sales data shown to determine the percent change for each of the months given.

b. Find the exponential function that models the data.c. Graph the data and the model on the same axes.

Example (cont)

a. Use the monthly sales data shown to determine the percent change for each of the months given.

Solution

The percent change of the outputs is approximately -22%. This means that the sales 1 month from now will be approximately 22% less than the sales now.

Example (cont)

b. Find the exponential function that models the data. Solutionb. Because the percent change is nearly constant, an exponential function should fit these data well. Technology gives the model y = 999.781(0.780 x) where x is the month and y is the sales in thousands of dollars.c. Graph the data and the model on the same axes.

Example

Suppose inflation averages 6% per year for each year from 2000 to 2010. This means that an item that costs $1 one year will cost $1.06 one year later. In the second year, the $1.06 cost will increase by a factor of 1.06, to (1.06)(1.06) = 1.062.a. Write an expression that gives the cost t years after 2000 of an item costing $1 in 2000.b. Write an exponential function that models the cost of an item t years from 2000 if its cost was $100 in 2000.c. Use the model to find the cost of the item from part (b) in 2010.

Example (cont)

a. Write an expression that gives the cost t years after 2000 of an item costing $1 in 2000. Solutiona. The cost of an item is $1 in 2000, and the cost increases at a constant percent of 6% = 0.06 per year, so the cost after t years will be 1(1 + 0.06)t = 1.06t dollars.

b. Write an exponential function that models the cost of an item t years from 2000 if its cost was $100 in 2000.Solutionb. The inflation rate is 6% = 0.06. Thus, if an item costs $100 in 2000, the function that gives the cost after t years is f(t) = 100(1 + 0.06)t = 100(1.06t ) dollars.

Example (cont)

c. Use the model to find the cost of the item from part (b) in 2010.SolutionThe year 2010 is 10 years after 2000, so the cost of an item costing $100 in 2000 isf(10) = 100(1.0610) = 179.08 dollars.

Example

As the table shows, projections indicate that the percent of U.S. adults with diabetes could dramatically increase.

a. Find a logarithmic model that fits the data, with x = 0 in 2000.b. Use the reported model to predict the percent of U.S. adults with diabetes in 2027.c. In what year does this model predict the percent to be 26.9%?

Example (cont)

a. Find a logarithmic model that fits the data, with x = 0 in 2000.

SolutionWe use logarithmic regression on a graphing utility to find the function that models the data. This function, rounded to three decimal places, is y = –12.975 + 11.851 ln x where x is the number of years after 2000.

Example (cont)

b. Use the reported model to predict the percent of U.S. adults with diabetes in 2027.SolutionEvaluating the reported model at x = 27 gives y = –12.975 + 11.851 ln 27 ≈ 26.1 percent in 2027.

Example (cont)

c. In what year does this model predict the percent to be 26.9%?

Solutionc. Setting y = 26.9 gives

26.9 = –12.975 + 11.851 ln x 39.875 = 11.851 ln x 3.3647 = ln x x = e3.3647 = 28.9

We could also solve graphically by intersecting the graphs of y1 = –12.975 + 11.851 ln x and y2 = 26.9, Thus, the percent reaches 27% in 2029.