Exam 5: Exponential and Logarithmic Functions

The president of a company predicts that sales N will increase after she assumes office and that the number of monthly sales will follow the curve given by , where t represents time in months since she assumed office. What will the sales be when she assumes office? ​

The following figure, from Investor's Business Daily (March 5, 1998), shows how quickly the U.S. metal processing industry isconsolidating. The linear equation that is the best fit for the number of metal processors as a function of years after 1990 is , and the best exponential fit is . The linear equation seems to give a much better fit for the data points than the exponential equation. Why then is the exponential equation a more useful model to predict the number of metal processors in ? ​ ​

Solve the logarithmic equation . Round your answer to three decimal places. ​

If the total cost function for a product is , where x is the number of items produced, what is the total cost of producing 50 units? Round your answer to the nearest cent. ​

Graph the function. ​ ​

Solve the exponential equation. Give answers correct to 3 decimal places. ​ ​

Graph the function. ​ ​

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 year when the number of mutual funds will reach 8,000. ​

Solve the exponential equation. Give the answer correct to 3 decimal places. ​ ​

Evaluate the logarithm by using properties of logarithms and the following facts. Round your answer two decimal places. ​ ​ ​ ​

By using data from the U.S. Bureau of Labor Statistics for the years 1968-2002, the purchasing power P of a 1983 dollar can be modeled with the function , where t is the number of years past 1960. In what year will the purchasing power of a 1983 dollar be $0.20? Round-up your answer to the nearest year. ​

Use a change-of-base formula to evaluate with a calculator or other technology. Round your answer to four decimal places. ​

Use properties of logarithms or a definition to simplify the expression. Check the result with a change-of-base formula and a calculator. Round your answer two decimal places. ​

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. Find an equation that models these data and use it to predict the consumer price index in 2015. Use the model to predict the consumer price index in 2015. Round your answer to two decimal places. ​ ​ Source: U.S. Bureau of the Census ​

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

If the supply function for a product is given by , where q represents the number of hundreds of units, what will be the price when the producers are willing to supply 200 units? Round your answer to the nearest cent. ​

If the demand function for a product is given by where p is the price per unit when x units are demanded, what is the total revenue when 90 units are demanded and supplied? Round your answer to the nearest cent. ​

Is the function a growth exponential or decay exponential? ​

The securities industry experienced dramatic growth in the last two decades of the 20th century. The following models for the industry's revenue R and expenses or costs C (both in billions of dollars) were developed as functions of the years past 1980 with data from the U.S. Securities and Exchange Commission's 2000 Annual Report (2001). and . Use the models to predict the profit for the securities industry in 2009. Round your answer to two decimal places. ​

The purchasing power P (in dollars) of an annual amount of A dollars after t years of 4% inflation decays according to . How long will it be before a pension of $60,000 per year has a purchasing power of $30,000? Round your answer to one decimal place. ​