Exam 5: Exponential and Logarithmic Functions

The consumer price index (CPI) is calculated by averaging the prices of various items after assigning a weight to each item. The following table gives the consumer price indexes for selected years from 1940 through 2002, reflecting buying patterns of all urban consumers, with x representing years past 1900. Find an equation that models these data. If it's necessary, round your calculations to four decimal places. ​ ​ Source: U.S. Bureau of the Census ​

The demand function for a certain commodity is given by , where q is number of units. At what price per unit will the quantity demanded equal 6 units? Round your answer to the nearest cent. ​

For selected years from 1960 to 2001, the national health care expenditures H, in billions of dollars, can be modeled by , where t is the number of years past 1960 (Source: U.S. Department of Health and Human Services). If this model remains accurate, in what year will national health care expenditures reach $3 trillion (that is, $3000 billion)? Round your answer to the nearest year. ​

Total personal income in the United States (in billions of dollars) for selected years from 1960 to 2002 is given in the following table. ​ Source: Bureau of Economic Analysis, U.S. Department of Commerce These data can be modeled by an exponential function. Write the equation of this function, with x as the number of years past 1960. If it's necessary round your calculations to four decimal places. ​

Use a calculator to evaluate the expression. Round your answer to six decimal places. ​ ​

The monthly sales S for a product is given by , where x is the number of months that have passed since the end of a promotional campaign. Determine the monthly sales 4 months after the promotional campaign. Round your answer to the nearest cent. ​

Write the expression as the sum or difference of two logarithmic functions containing no exponents. ​ ​

Suppose that the equation represents the number of employees working t years after a company begins operations. After how many years will at least 300 employees be working? Round your answer to two decimal places. ​

The following table gives the average number of students per computer in public schools for the school years that ended in 1985 through 2002. Let x be the number of years past 1980. Find an exponential model for these data. Round your answer to three decimal places. ​ Source: Quality Education Data, Inc., Denver, Colorado ​

Graph the function. ​ ​

Let . Using a graphing utility, graph for ​ and . What effect does c have on the graphs? ​

Pollution levels in Lake Erie have been modeled by the equation , where x is the volume of pollutants (in cubic kilometers) and t is the time (in years) (Adapted from R. H. Rainey, Science 155 (1967), 1242-1243). Find the initial pollution level; that is, find x when t=0. ​

If $7,000 is invested at an annual rate of 11.5% compounded continuously, the future value S at any time t (in years) is given by . What is the amount after 12 months? Round your answer to the nearest cent. ​

Solve for x by writing the equation in exponential form. ​ ​

Solve for x by writing the equation in exponential form. Round your answer two decimal places, if necessary. ​ ​

A starfish population can be modeled by , where is the number of individuals at time , r is the yearly rate of growth, and t is the number of years. Sketch the graph for to when the growth rate is 2.1% and is 5.5 billion. All numbers on the vertical axis are in billions. ​

For selected years from 1978 to 2002, the number of mutual funds N, excluding money market funds, can be modeled by , where t is the number of years past 1975 (Source: Investment Company Institute, Mutual Fund Fact Book (2003)). Use the model to estimate the number of mutual funds in 2004. Round your answer to the nearest whole number. ​

Let . Use a graphing utility to graph the functions and where . Identify the graphs of and below. ​

Use the formula . ​ The San Francisco earthquake of 1906 measured 8.25 on the Richter scale, and the San Francisco earthquake of 1989 measured 7.1. How much more intense was the 1906 quake? ​

Use a calculator to evaluate the expression. Round your answer to two decimal places, if necessary. ​