Chapter 9

Write each equation in logarithmic form. \(3^{3}=27\)

Solve each equation or inequality. Round to four decimal places. $$ 2^{3 p}>1000 $$

Write an equivalent exponential or logarithmic equation. \(\ln \frac{7}{3}=2 x\)

Solve for \(n\) \(\log _{a}(4 n)-2 \log _{a} x=\log _{a} x\)

Write each equation in logarithmic form. \(5^{-3}=\frac{1}{125}\)

Solve each equation or inequality. Round to four decimal places. $$ \log _{b} 81=2 $$

Write an equivalent exponential or logarithmic equation. \(\ln e^{x}=3\)

Solve for \(n\) \(\log _{b} 8+3 \log _{b} n=3 \log _{b}(x-1)\)

Write each equation in logarithmic form. \(\left(\frac{1}{3}\right)^{-2}=9\)

For Exercises \(33-35,\) use the following information. A small corporation decides that 8\(\%\) of its profits would be divided among its six managers. There are two sales managers and four nonsales managers. Fifty percent would be split equally among all six managers. The other 50\(\%\) would be split among the four nonsales managers. Let \(p\) represent the profits. Write an expression to represent the share of the profits each nonsales manager will receive.

Solve each equation. Round to the nearest ten-thousandth. \(3 e^{x}+1=5\)

Solve each equation. Check your solutions. \(\log _{10} z+\log _{10}(z+3)=1\)

Solve each equation or inequality. Round to four decimal places. $$ 3^{y+2} \geq 8^{3 y} $$

Write each equation in logarithmic form. \(100^{\frac{1}{2}}=10\)

Write an exponential function for the graph that passes through the given points. $$ (0,-0.3) \text { and }(5,-9.6) $$

Solve each equation. Check your solutions. \(\log _{6}\left(a^{2}+2\right)+\log _{6} 2=2\)

Solve each equation. Round to the nearest ten-thousandth. \(2 e^{x}-1=0\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{2} 13 $$

Write each equation in logarithmic form. \(2401^{\frac{1}{4}}=7\)

For Exercises \(33-35,\) use the following information. A small corporation decides that 8\(\%\) of its profits would be divided among its six managers. There are two sales managers and four nonsales managers. Fifty percent would be split equally among all six managers. The other 50\(\%\) would be split among the four nonsales managers. Let \(p\) represent the profits. Write an expression in simplest form to represent the share of the profits each sales manager will receive.

Solve each equation. Check your solutions. \(\log _{2}(12 b-21)-\log _{2}\left(b^{2}-3\right)=2\)

Solve each equation. Round to the nearest ten-thousandth. \(-3 e^{4 x}+11=2\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{5} 20 $$

Evaluate each expression. \(\log _{2} 16\)

For Exercises \(36-38,\) use the graph at the right. U.S. growers were forecasted to produce 264 million pounds of pecans in 2003 . Write the number of pounds of pecans forecasted by U.S. growers in 2003 in scientific notation.

Solve each equation. Check your solutions. \(\log _{2}(y+2)-\log _{2}(y-2)=1\)

Solve each equation. Round to the nearest ten-thousandth. \(8+3 e^{3 x}=26\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{7} 3 $$

Evaluate each expression. \(\log _{12} 144\)

Exercises \(36-38\) , use the following information. Suppose you deposit a principal amount of \(P\) dollars in a bank account that pays compound interest. If the annual interest rate is \(r\) (expressed as a decimal) and the bank makes interest payments \(n\) times every year, the amount of money \(A\) you would have after \(t\) years is given by \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). If the principal, interest rate, and number of interest payments are known, what type of function is \(A(t)=P\left(1+\frac{r}{n}\right)^{n t} ?\) Explain your reasoning.

Solve each equation. Check your solutions. \(\log _{3} 0.1+2 \log _{3} x=\log _{3} 2+\log _{3} 5\)

Solve each equation. Round to the nearest ten-thousandth. \(2 e^{x}-3=-1\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{3} 8 $$

Evaluate each expression. \(\log _{16} 4\)

Exercises \(36-38\) , use the following information. Suppose you deposit a principal amount of \(P\) dollars in a bank account that pays compound interest. If the annual interest rate is \(r\) (expressed as a decimal) and the bank makes interest payments \(n\) times every year, the amount of money \(A\) you would have after \(t\) years is given by \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Write an equation giving the amount of money you would have after t years if you deposit \(\$ 1000\) into an account paying 4\(\%\) annual interest compounded quarterly (four times per year).

Solve each equation. Check your solutions. \(\log _{5} 64-\log _{5} \frac{8}{3}+\log _{5} 2=\log _{5}(4 p)\)

Solve each equation. Round to the nearest ten-thousandth. \(-2 e^{x}+3=0\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{4}(1.6)^{2} $$

Evaluate each expression. \(\log _{9} 243\)

Solve each equation. Round to the nearest ten-thousandth. \(-2+3 e^{3 x}=7\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{6} \sqrt{5} $$

Evaluate each expression. \(\log _{2} \frac{1}{32}\)

Solve each equation. Check your solution. $$ 2^{3 x+5}=128 $$

Solve each equation. Round to the nearest ten-thousandth. \(1-\frac{1}{3} e^{5 x}=-5\)

Evaluate each expression. \(\log _{3} \frac{1}{81}\)

Solve each equation. Check your solution. $$ 5^{n-3}=\frac{1}{25} $$

In 2005 , the world's population was about 6.5 billion. If the world's population continues to grow at a constant rate, the future population \(P,\) in billions, can be predicted by \(P=6.5 e^{0.02 t},\) where \(t\) is the time in years since 2005. According to this model, what will the world’s population be in 2015?

Evaluate each expression. \(\log _{10} 0.001\)

Solve each equation. Check your solution. $$ \left(\frac{1}{9}\right)^{m}=81^{m+4} $$

In 2005 , the world's population was about 6.5 billion. If the world's population continues to grow at a constant rate, the future population \(P,\) in billions, can be predicted by \(P=6.5 e^{0.02 t},\) where \(t\) is the time in years since 2005. Some experts have estimated that the world’s food supply can support a population of at most 18 billion. According to this model, for how many more years will the food supply be able to support the trend in world population growth?