91Ó°ÊÓ

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function. $$ y=0.25 e^{-3 x} $$

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function. $$ y=0.4 e^{-0.25 x} $$

The value of a mountain bike \(y\) (in dollars) can be approximated by the model \(y=200(0.75)^t\), where \(t\) is the number of years since the bike was new. (See Example 2.) a. Tell whether the model represents exponential growth or exponential decay. b. Identify the annual percent increase or decrease in the value of the bike. c. Estimate when the value of the bike will be \(\$ 50\).

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function. $$ y=0.6 e^{0.5 x} $$

The population \(P\) (in thousands) of Austin, Texas, during a recent decade can be approximated by \(y=494.29(1.03)^t\), where \(t\) is the number of years since the beginning of the decade. a. Tell whether the model represents exponential growth or exponential decay. b. Identify the annual percent increase or decrease in population. c. Estimate when the population was about 590,000 .

In 2006, there were approximately 233 million cell phone subscribers in the United States. During the next 4 years, the number of cell phone subscribers increased by about \(6 \%\) each year. (See Example 3.) a. Write an exponential growth model giving the number of cell phone subscribers \(y\) (in millions) \(t\) years after 2006 . Estimate the number of cell phone subscribers in 2008. b. Estimate the year when the number of cell phone subscribers was about 278 million.

You take a 325 milligram dosage of ibuprofen. During each subsequent hour, the amount of medication in your bloodstream decreases by about \(29 \%\) each hour. a. Write an exponential decay model giving the amount \(y\) (in milligrams) of ibuprofen in your bloodstream \(t\) hours after the initial dose. b. Estimate how long it takes for you to have 100 milligrams of ibuprofen in your bloodstream.

Justify each step in rewriting the exponential function. \(\begin{aligned} y &=a(3)^{t / 14} \\ &=a\left[(3)^{1 / 14}\right]^t \\ & \approx a(1.0816)^t \\\ &=a(1+0.0816)^t \end{aligned}\)

WRITING Explain why the expressions \(\log _2(-1)\) and \(\log _1 1\) are not defined.

In Exercises 27-30, use the properties of exponents to rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then find the percent rate of change. $$ y=e^{-0.25 t} $$