Exam 5: Exponential and Logarithmic Functions

Use the following information. Chemists use the pH (hydrogen potential) of a solution to measure its acidity or basicity. The pH is given by the formula , where is the concentration of hydrogen ions in moles per liter. What value of is associated with pH level 12? ​​

Suppose that the market share y (as a percent) that a company expects t months after a new product is introduced is given by . How long (to the nearest month) will it take for the market share to be 25%? Round your answer to one decimal place. ​

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

Graph the function. ​ ​

If $4,500 is invested at an annual rate of 8% compounded continuously, the future value S at any time t (in years) is given by . How long does it take for the investment to double? Round your answer to one decimal place. ​

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

Write the equation in exponential form. ​ ​

Between the years 1960 and 2002, the percent of women in the work force is given by , where x is the number of years past 1950 (Source: U.S. Bureau of Labor Statistics). Graph this function with a graphing utility and use the graph drawn to estimate the year in which the percent reached 50. Round your answer to the nearest year. ​

Use properties of logarithms or a definition to simplify the expression. Check the result with a change-of-base formula and a calculator. ​ ​

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

Use properties of logarithms or a definition to simplify the expression. ​ If find . ​

The purchasing power P (in dollars) of an annual amount of A dollars after t years of 4% inflation decays according to . Determine how large a pension A needs to be so that the purchasing power P is $70,000 after 25 years? Round your answer to the nearest dollar. ​

An initial amount of 300 g of the radioactive isotope thorium-234 decays according to , where t is in years. How long does it take for half of the initial amount to disintegrate? This time is called the half-life of this isotope. Round your answer to one decimal place. ​

​We will show in the next chapter that if $P is invested for n years at an annual rate of 3% compounded continuously, the future value of the investment is given by ​ ​ Use and graph this function for . ​

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. How many months after the end of the campaign will sales drop below 6,000, if no new campaign is initiated? Round to two decimal places. ​

Express in the form with an appropriate value of . ​

The following table gives the average number of students per computer in public schools for the school years that ended in 1985 through 2002. Find an exponential model for these data. How many students per computer in public schools does this model predict for 2005?​ Round your answer to two decimal places. ​​​

The following table gives the percent of the U.S. population with Internet connections for the years 1997 to 2003. Use a calculator to find the logistic function that models these data. Use x as the number of years past 1995. ​ ​ Source: U.S. Department of Commerce ​

Use the properties of logarithms to write the expression as a single logarithm. ​ ​

Between the years 1960 and 2002, the percent of women in the work force is given by , where x is the number of years past 1950 (Source: U.S. Bureau of Labor Statistics). What does this model estimate to be the percent of women in the work force in 2019? Round your answer to two decimal places. ​