Solving Logarithmic Equations (Word Problems)

1 Solving Logarithmic Equations ( word problems ) Example 1 INVESTMENT Mr. and Mrs. Mitchell are saving for their daughter s college education. They invest $10,000 in an account that pays interest compounded continuously with the goal to have twice that amount in the account in ten years. a. Will the Mitchell s reach their investment goal in ten years? Find the doubling time for the Mr. and Mrs. Mitchell s investment. For continuously compounded interest, the constant k is the interest rate, written as a decimal. t = ln 2k = ln The decimal for is years Use a calculator. Ten years is not enough time for their investment to double. b. What interest rate is required for an investment with continuously compounded interest to double in 10 years? t = ln 2k 10 = ln 2k k = ln 210 solve for k. k years An interest rate of is required for an investment with continuously compounded interest to double in 10 years.

2 Example 2 TELECOMMUNICATIONS The table below gives the number of cellular telephone subscriptions from 1990 to 2001. Year 1990 1991 1992 1993 1994 1995 Years since 1990 0 1 2 3 4 5 Subscribers (in millions) Year 1996 1997 1998 1999 2000 2001 Years since 1990 6 7 8 9 10 11 Subscribers (in millions) a. Find a function that models the subscription data shown. Enter the data on the STAT EDIT screen. In order to work with smaller numbers, use the number of years since 1990 as the independent variable. Then draw a scatter plot. The scatter plot suggests an exponential model. To have the calculator find a regression equation of the form y = abx, use ExpReg from the STAT CALC screen. The equation of an exponential function that models the subscription data is y = ( )x. The equation can be written in terms of base e as follows.

3 Y = ( )x y = (eln )x eln a = a y = (ln )x (am)n = amn y = ln b. Use the equation to predict the number of cellular subscribers in 2020. To predict the number of cellular subscriptions in 2020, note that 2020 is 30 years after 1990. y = ( )x = ( )30 Replace x with 30. = 41, The model predicts that there will 41, million or 41,283,450,000 cellular subscriptions in 2020. Example 3 METEOROLOGY The table below shows what the wind chill would be on a 30-degree day at various wind speeds. Find an equation to model the data. Wind Speed (mph) 5 10 15 20 25 30 35 40 Wind Chill 25 21 19 17 16 15 14 13 Make a scatter plot of the data. Let the independent variable be wind speed and let wind chill be the dependent variable. Since the scatter plot does not appear to flatten out as much as a decaying exponential function would, we will look for a Logarithmic model of the form y = a + b ln x where b < 0 for the data.

4 To have the calculator find a regression equation of the form y = a + b ln x, use LnReg from the STAT CALC screen. The equation of a Logarithmic function that models the data is y = - ln x. Example 4 The table below shows the number of web sites on the Internet from 1998 to 2004. Year 1998 1999 2000 2001 2002 2003 2004 Web Sites (billions) 50 80 190 330 590 900 1400 a. Linearize the data. That is, make a table with x- and y-values, where x is the number of years since 1998 and y is the number of web sites. Then make a scatter plot of the linearized data. Subtract 1998 from each year and find the natural logarithm of each web site quantity. x 0 1 2 3 4 5 6 ln y The scatter plot suggests that there may be a linear relationship between x and ln y. b. Find a regression equation for the linearized data.

5 Use LinReg(ax+b) on the STAT CALC screen to find the linear regression equation. We can use the equation ln y = + to model ln y. c. Use the linear regression equation to find an exponential model for the original data. To find a model solve the regression equation in part b for y. ln y = + eln y = + Raise e to each side. y = + eln y = y y = Product Property of Exponents y = The number of web sites on the Internet between 1998 and 2004 can be modeled by the exponential function y = d. Use the exponential model to predict the number of web sites that there will be in 2015. The year 2015 is 17 years after 1998, so replace x with 17 in the exponential function. y = y = (17) or about 861, billion The model predicts 861, billion or trillion web sites in 2015.