Chapter 4

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$f(x)=5^{-x}$$

Determine whether each \(x\)-value is a solution of the equation. \(\log _{6}\left(\frac{5}{3} x\right)=2\) (a) \(x \approx 20.2882\) (b) \(x=\frac{108}{5}\) (c) \(x=21.6\)

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{16} 8=\frac{3}{4}$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$h(x)=10^{-x}$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,2.0),(1.5,3.5),(2,4.0),(4,5.8),(6,7.0),(8,7.8)$$

Determine whether each \(x\)-value is a solution of the equation. \(\ln (x-1)=3.8\) (a) \(x=1+e^{3.8}\) (b) \(x \approx 45.7012\) (c) \(x=1+\ln 3.8\)

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{3} 7$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{2} \sqrt{2}=\frac{1}{2}$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$h(x)=\left(\frac{5}{4}\right)^{x}$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,5.8),(1.5,6.0),(2,6.5),(4,7.6),(6,8.9),(8,10.0)$$

Determine whether each \(x\)-value is a solution of the equation. \(\ln (2+x)=2.5\) (a) \(x=e^{2.5}-2\) (b) \(x \approx \frac{4073}{400}\) (c) \(x=\frac{1}{2}\)

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{9} 4$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{5} \sqrt[3]{25}=\frac{2}{3}$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$g(x)=\left(\frac{3}{2}\right)^{x}$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,4.4),(1.5,4.7),(2,5.5),(4,9.9),(6,18.1),(8,33.0)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=64^{x}\\\&g(x)=8\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{1 / 2} 16$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$5^{3}=125$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$g(x)=\left(\frac{5}{4}\right)^{-x}$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,11.0),(1.5,9.6),(2,8.2),(4,4.5),(6,2.5),(8,1.4)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=27^{x}\\\&g(x)=9\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{1 / 8} 64$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$8^{2}=64$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$f(x)=\left(\frac{3}{2}\right)^{-x}$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,7.5),(1.5,7.0),(2,6.8),(4,5.0),(6,3.5),(8,2.0)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=5^{x-2}-15\\\&g(x)=10\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{6} 0.9$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$81^{1 / 4}=3$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,5.0),(1.5,6.0),(2,6.4),(4,7.8),(6,8.6),(8,9.0)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=2^{-x+1}-3\\\&g(x)=13\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{4} 0.045$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$9^{3 / 2}=27$$

Use the graph of \(y=2^{x}\) to match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] $$f(x)=2^{-x}$$

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0,5),(1,6),(2,7),(3,9),(4,13)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=4 \log _{3} x\\\&g(x)=20\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{15} 1460$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$6^{-2}=\frac{1}{36}$$

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0,4.0),(2,6.9),(4,18.0),(6,32.3),(8,59.1),(10,118.5)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=3 \log _{5} x\\\&g(x)=6\end{aligned}$$

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{20} 175$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$10^{-3}=0.001$$

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0,8.3),(1,6.1),(2,4.6),(3,3.8),(4,3.6)$$

Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple when interest is compounded continuously at rate \(r .\) Create a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array}$$

Rewrite the expression in terms of \(\ln 4\) and \(\ln 5 .\),$$\ln 20$$.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=\ln e^{x+1}\\\&g(x)=2 x+5\end{aligned}$$

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$g^{a}=4$$

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(-3,102.2),(0,80.5),(3,67.8),(6,58.2),(10,55.0)$$

Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple when interest is compounded annually at rate \(r .\) Create a scatter plot of the data. $$\begin{array}{|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array}$$

Rewrite the expression in terms of \(\ln 4\) and \(\ln 5 .\), $$\ln 500$$.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=\ln e^{x-2}\\\&g(x)=3 x+2\end{aligned}$$