Chapter 12

Find the exact value of each logarithm. $$ \ln e^{4} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=-4 x ; g(x)=x^{3}+x^{2}-6 $$

Graph each exponential function. $$ G(x)=3^{x-2} $$

Use the power property to rewrite each expression. $$ \log _{6} 7^{-2} $$

Solve each equation. $$ \log _{2}\left(x^{2}+x\right)=1 $$

Write each as a logarithmic equation. $$ e^{3}=x $$

Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 40 & 7 & 14 & & & \\ \hline \text { b. } 40 & 7 & 11 & & & \\ \hline \end{array} $$

Find the exact value of each logarithm. $$ \ln \sqrt[4]{e} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=|x| ; g(x)=10 x-3 $$

Solve. $$ 3^{x}=27 $$

Use the power property to rewrite each expression. $$ \log _{5} \sqrt{y} $$

Solve each equation. $$ \log _{6}\left(x^{2}-x\right)=1 $$

Write each as a logarithmic equation. $$ e^{5}=y $$

Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 200 & 12 & 36 & & & \\ \hline \text { b. } 200 & 12 & 40 & & & \\ \hline \end{array} $$

Find the exact value of each logarithm. $$ \ln \sqrt[5]{e} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=|x| ; g(x)=14 x-8 $$

Solve. $$ 6^{x}=36 $$

Use the power property to rewrite each expression. $$ \log _{5} \sqrt[3]{x} $$

Solve each equation. $$ \log _{4} x+\log _{4}(x+6)=2 $$

Write each as a logarithmic equation. $$ 10^{-1}=\frac{1}{10} $$

Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 21 & 152 & 500 & & & \\ \hline \end{array} $$

Find the exact value of each logarithm. $$ \log 10^{3} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x+4 $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=\sqrt{x} ; g(x)=-5 x+2 $$

Solve. $$ 16^{x}=8 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{2} 5+\log _{2} x^{3} $$

Solve each equation. $$ \log _{3} x+\log _{3}(x+6)=3 $$

Write each as a logarithmic equation. $$ 10^{-2}=\frac{1}{100} $$

Find the exact value of each logarithm. $$ \ln e^{5} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x-5 $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=7 x-1 ; g(x)=\sqrt[3]{x} $$

Solve. $$ 64^{x}=16 $$

Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 35 & 119 & 500 & & & \\ \hline \end{array} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 2+\log _{5} y^{2} $$

Solve each equation. $$ \log _{5}(x+3)-\log _{5} x=2 $$

Write each as a logarithmic equation. $$ 4^{-2}=\frac{1}{16} $$

Solve. Round answers to the nearest tenth. A form of nickel has a half-life of 96 years. How much of a 30 -gram sample is left after 250 years?

Find the exact value of each logarithm. $$ \ln e^{3.1} $$

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ H(x)=\sqrt{x^{2}+2} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=2 x-3 $$

Solve. $$ 32^{2 x-3}=2 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 3 \log _{4} 2+\log _{4} 6 $$

Solve each equation. $$ \log _{6}(x+2)-\log _{6} x=2 $$

Write each as a logarithmic equation. $$ 3^{-4}=\frac{1}{81} $$

Solve. Round answers to the nearest tenth. A form of uranium has a half-life of 72 years. How much of a 100 -gram sample is left after 500 years?

Find the exact value of each logarithm. $$ \log 10^{7} $$

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ G(x)=\sqrt{3 x} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=4 x+9 $$

Solve. $$ 9^{2 x+1}=81 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{3} 5+\log _{3} 2 $$