Chapter 12

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 305 & 5 \% & 8 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \ln 0.0716 $$

For the functions \(f\) and \(g\), find a. \((f+g)(x)\), b. \((f-g)(x), c .(f \cdot g)(x)\), and d. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=-3 x ; g(x)=5 x^{2} $$

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Month of } 2009 \text { (Input) } & \text { July } & \text { August } & \text { September } & \text { October } & \text { November } & \text { December } \\ \hline \text { Unemployment Rate in Percent (0utput) } & 9.4 & 9.7 & 9.8 & 10.1 & 10.0 & 10.0 \\ \hline \end{array} $$

Graph each exponential function. $$ y=\left(\frac{1}{2}\right)^{x}-2 $$

Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{10} 5+\log _{10} 2+\log _{10}\left(x^{2}+2\right) $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ e^{2 x}=8 $$

Write each as an exponential equation. $$ \log _{e} y=7 $$

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 402 & 7 \% & 5 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \ln 0.0032 $$

For the functions \(f\) and \(g\), find a. \((f+g)(x)\), b. \((f-g)(x), c .(f \cdot g)(x)\), and d. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=4 x^{3} ; g(x)=-6 x $$

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State (Input) } & \text { Texas } & \text { Massachusetts } & \text { Nevada } & \text { Idaho } & \text { Wisconsin } \\ \hline \text { Number of Two-Year Colleges (0utput) } & 70 & 22 & 3 & 3 & 31 \\\ \hline \end{array} $$

Graph each exponential function. $$ y=\left(\frac{1}{3}\right)^{x}+2 $$

Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{6} 3+\log _{6}(x+4)+\log _{6} 5 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 2^{x-3}=5 $$

Write each as an exponential equation. $$ \log _{e} \frac{1}{e^{2}}=-2 $$

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 10,000 & 12 \% & 15 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \log 12.6 $$

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (f \circ g)(2) $$

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { State (Input) } & \text { California } & \text { Alaska } & \text { Indiana } & \text { Louisiana } & \text { New Mexico } & \text { Ohio } \\ \hline \text { Rank in Population (Output) } & 1 & 47 & 16 & 25 & 36 & 7 \\ \hline \end{array} $$

Graph each exponential function. $$ y=-2^{x} $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 12-\log _{5} 4 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 8^{x-2}=12 $$

Write each as an exponential equation. $$ \log _{e} \frac{1}{e}=-1 $$

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 15,000 & 16 \% & 11 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \log 25.9 $$

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (h \circ f)(-2) $$

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Shape (Input) } & \text { Triangle } & \text { Pentagon } & \text { Quadrilateral } & \text { Hexagon } & \text { Decagon } \\ \hline \text { Number of Sides (Output) } & 3 & 5 & 4 & 6 & 10 \\ \hline \end{array} $$

Graph each exponential function. $$ y=-3^{x} $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{7} 20-\log _{7} 4 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 4^{x+7}=3 $$

Write each as an exponential equation. $$ \log _{7} \sqrt{7}=\frac{1}{2} $$

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 207,000 & 32 \% & 25 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \ln 5 $$

Given the one-to-one function \(f(x)=x^{3}+2,\) find the following. (Hint: You do not need to find the equation for \(f^{-1}\).) a. \(f(1)\) b. \(f^{-1}(3)\)

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (g \circ f)(-1) $$

Graph each exponential function. $$ y=3^{x}-2 $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{3} 8-\log _{3} 2 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 6^{x+3}=2 $$

Write each as an exponential equation. $$ \log _{11} \sqrt[4]{11}=\frac{1}{4} $$

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 325,000 & 29 \% & 31 & \\ \hline \end{array} $$

Use a calculator to approximate each logarithm to four decimal places. $$ \ln 7 $$

Given the one-to-one function \(f(x)=x^{3}+2,\) find the following. (Hint: You do not need to find the equation for \(f^{-1}\).) a. \(f(0)\) b. \(f^{-1}(2)\)

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (f \circ h)(1) $$

Graph each exponential function. $$ y=2^{x}-3 $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 12-\log _{5} 3 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 7^{3 x-4}=11 $$

Write each as an exponential equation. $$ \log _{0.7} 0.343=3 $$

Solve. Unless noted otherwise, round answers to the nearest whole. Suppose a city with population 500,000 has been growing at a rate of \(3 \%\) per year. If this rate continues, find the population of this city in 12 years.

Use a calculator to approximate each logarithm to four decimal places. $$ \log 41.5 $$