Chapter 9

POLICE Police use blood alcohol content (BAC) to measure the percent concentration of alcohol in a person's bloodstream. In most states, a BAC of 0.08 percent means a person is not allowed to drive. Each hour after drinking, a person's BAC may decrease by 15\(\% .\) If a person has a BAC of 0.18 , how many hours will he need to wait until he can legally drive?

Use a calculator to evaluate each expression to four decimal places. \(e^{6}\)

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} 18\)

Write each equation in logarithmic form. \(5^{4}=625\)

For Exercises \(2-4,\) use the following information. A radioisotope is used as a power source for a satellite. The power output \(P\) (in watts) is given by \(P=50 e^{-\frac{t}{250}},\) where \(t\) is the time in days. Is the formula for power output an example of exponential growth or decay? Explain your reasoning.

Use a calculator to evaluate each expression to four decimal places. \(e^{-3.4}\)

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} 14\)

Write each equation in logarithmic form. \(7^{-2}=\frac{1}{49}\)

For Exercises \(2-4,\) use the following information. A radioisotope is used as a power source for a satellite. The power output \(P\) (in watts) is given by \(P=50 e^{-\frac{t}{250}},\) where \(t\) is the time in days. Find the power available after 100 days.

Use a calculator to evaluate each expression to four decimal places. \(e^{0.35}\)

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} \frac{7}{2}\)

Write each equation in logarithmic form. \(3^{5}=243\)

For Exercises \(2-4,\) use the following information. A radioisotope is used as a power source for a satellite. The power output \(P\) (in watts) is given by \(P=50 e^{-\frac{t}{250}},\) where \(t\) is the time in days. Ten watts of power are required to operate the equipment in the satellite. How long can the satellite continue to operate?

Use a calculator to evaluate each expression to four decimal places. \(\ln 1.2\)

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} \frac{2}{3}\)

Write each equation in exponential form. \(\log _{3} 81=4\)

Sketch the graph of each function. Then state the function's domain and range. $$ y=3(4)^{x} $$

STANDARDIZED TEST PRACTICE The weight of a bar of soap decreases by 2.5\(\%\) each time it is used. If the bar weighs 95 grams when it is new, what is its weight to the nearest gram after 15 uses? $$\begin{array}{llll}{\text { A } 57.5 \mathrm{g}} & {\text { B } 59.4 \mathrm{g}} & {\text { C } 65 \mathrm{g}} & {\text { D } 93 \mathrm{g}}\end{array}$$

Use a calculator to evaluate each expression to four decimal places. \(\ln 0.1\)

Solve each equation. Round to four decimal places. $$ 9^{x}=45 $$

Write each equation in exponential form. \(\log _{36} 6=\frac{1}{2}\)

Sketch the graph of each function. Then state the function's domain and range. $$ y=2\left(\frac{1}{3}\right)^{x} $$

For Exercises 6 and \(7,\) use the following information. Fayette County, Kentucky, grew from a population of \(260,512\) in 2000 to a population of \(268,080\) in \(2005 .\) Write an exponential growth equation of the form \(y=a e^{k t}\) for Fayette County, where \(t\) is the number of years after 2000 .

Use a calculator to evaluate each expression to four decimal places. \(\ln 3.25\)

Given \(\log _{2} 7 \approx 2.8074\) and \(\log _{5} 8 \approx 1.2920\) to approximate the value of each expression. \(\log _{2} 49\)

Solve each equation. Round to four decimal places. $$ 3.1^{a-3}=9.42 $$

Write each equation in exponential form. \(\log _{125} 5=\frac{1}{3}\)

Determine whether each function represents exponential growth or decay. $$ y=(0.5)^{x} $$

Write an equivalent exponential or logarithmic equation. 7\. \(e^{x}=4\)

Given \(\log _{2} 7 \approx 2.8074\) and \(\log _{5} 8 \approx 1.2920\) to approximate the value of each expression. \(\log _{5} 64\)

Solve each equation. Round to four decimal places. $$ 11^{x^{2}}=25.4 $$

Write each equation in exponential form. \(\log _{4} 256\)

Determine whether each function represents exponential growth or decay. $$ y=0.3(5)^{x} $$

Zeus Industries bought a computer for \(\$ 2500 .\) If it depreciates at a rate of 20\(\%\) per year, what will be its value in 2 years?

Write an equivalent exponential or logarithmic equation. \(\ln 1=0\)

Solve each equation. Check your solutions. \(\log _{3} 42-\log _{3} n=\log _{3} 7\)

Solve each equation. Round to four decimal places. $$ 7^{t-2}=5^{t} $$

Write each equation in exponential form. \(\log _{2} \frac{1}{8}\)

Write an exponential function for the graph that passes through the given points. $$ (0,3) \text { and }(-1,6) $$

A certain medication is eliminated from the bloodstream at a steady rate. It decays according to the equation \(y=a e^{-0.1625 t},\) where \(t\) is in hours. Find the half-life of this substance.

Solve each equation. Round to the nearest ten-thousandth. \(2 e^{x}-5=1\)

Solve each equation. Check your solutions. \(\log _{2}(3 x)+\log _{2} 5=\log _{2} 30\)

Solve each inequality. Round to four decimal places. $$ 4^{5 n}>30 $$

Write each equation in exponential form. \(\log _{6} 216\)

Write an exponential function for the graph that passes through the given points. $$ (0,-18) \text { and }(-2,-2) $$

Solve each equation. Round to the nearest ten-thousandth. \(3+e^{-2 x}=8\)

Solve each equation. Check your solutions. \(2 \log _{5} x=\log _{5} 9\)

Solve each inequality. Round to four decimal places. $$ 4^{p-1} \leq 3^{p} $$

Solve each equation. Check your solutions. \(\log _{9} x=\frac{3}{2}\)

MONEY For Exercises 10 and 11 , use the following information. In \(1993,\) My- Lien inherited \(\$ 1,000,000\) from her grandmother. She invested all of the money, and by 2005 , the amount had grown to \(\$ 1,678,000\) . Write an exponential function that could be used to model the money \(y .\) Write the function in terms of \(x,\) the number of years since \(1993 .\)