Chapter 4

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

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) 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. $$(1,2.0),(2,3.0),(3,3.5),(4,4.0),(5,4.1),(6,4.2),(7,4.5)$$

When \(\$ 1\) is invested in an account over a 10-year period, the amount \(A\) in the account after \(t\) years is given by \(A=1+0.075 t \quad\) or \(\quad A=e^{0.07 t}\) depending on whether the account pays simple interest at \(7 \frac{1}{2} \%\) or continuous compound interest at \(7 \% .\) Use a graphing utility to graph each function in the same viewing window. Which grows at a greater rate?

Rewrite the expression in terms of \(\ln 4\) and \(\ln 5 .\),$$\ln \frac{25}{4}$$.

Solve the exponential equation. $$4^{x}=16$$

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$f(x)=\log _{2} x$$

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) 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. $$(1, 8.5), (2, 11.4), (4, 12.8), (6, 13.6), (8, 14.2),(10, 14.6)$$

When \(\$ 1\) is invested in an account over a 10-year period, the amount \(A\) in the account after \(t\) years is given by \(A=1+0.06 t \quad\) or \(\quad A=\left(1+\frac{0.055}{365}\right)^{365 t}\) depending on whether the account pays simple interest at \(6 \%\) or compound interest at \(5 \frac{1}{2} \%\) compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a greater rate?

Solve the exponential equation. $$3^{x}=243$$

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$f(x)=\log _{16} x$$

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) 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. $$(1,11),(2,6),(3,5),(4,4),(5,3),(6,2)$$

Complete the table for the radioactive isotope. $$\begin{array}{lccc} & \text {Half-Life} & \text {Initial} & \text {Amount After} \\ \text {Isotope} & \text {(years)} & \text {Quantity} & \text {1000 Years} \\ ^{226} \mathrm{Ra} & 1600 & 10 \mathrm{g} & \end{array}$$

Approximate the logarithm using the properties of logarithms, given the values \(\log _{b} 2 \approx 0.3562\) \(\log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271 .\) Round your result to four decimal places.$$\log _{b} 8$$.

Solve the exponential equation. $$5^{x}=\frac{1}{625}$$

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$g(x)=\log _{10} x$$

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) 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. $$(3,14.6),(6,11.0),(9,9.0),(12,7.6),(15,6.5)$$

Approximate the logarithm using the properties of logarithms, given the values \(\log _{b} 2 \approx 0.3562\) \(\log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271 .\) Round your result to four decimal places.$$\log _{b} 30$$.

Solve the exponential equation. $$7^{x}=\frac{1}{49}$$

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$g(x)=\log _{10} x$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) 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. $$(1,2.0),(2,3.4),(5,6.7),(6,7.3),(10,12.0)$$

Approximate the logarithm using the properties of logarithms, given the values \(\log _{b} 2 \approx 0.3562\) \(\log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271 .\) Round your result to four decimal places.$$\log _{b} \frac{16}{25}$$.

Solve the exponential equation. $$\left(\frac{1}{8}\right)^{x}=64$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=14.8$$ $$x=4.3$$ (Function) $$f(x)=\log _{10} x$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) 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. $$(0.5, 1.0), (2, 12.5), (4, 33.2), (6, 65.7), (8, 98.5),(10, 150.0)$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=14.8$$ $$x=4.3$$ (Function) $$f(x)=\log _{10} x$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) 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. $$(1,10.0),(2,4.0),(3,0.7),(4,0.1)$$

Complete the table for the radioactive isotope. $$\begin{array}{lccc} & \text {Half-Life} & \text {Initial} & \text {Amount After} \\ \text {Isotope} & \text {(years)} & \text {Quantity} & \text {1000 Years} \\ ^{241} \mathrm{Am} & 432.2 & 26.4 \mathrm{g}& \end{array}$$

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{3}(x+1)$$

Solve the exponential equation. $$\left(\frac{2}{3}\right)^{x}=\frac{81}{16}$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=14.8$$ $$x=4.3$$ (Function) $$h(x)=6 \log _{10} x$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) 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. $$(2,450),(4,385),(6,345),(8,332),(10,312)$$

Complete the table for the radioactive isotope. $$\begin{array}{lccc} & \text {Half-Life} & \text {Initial} & \text {Amount After} \\ \text {Isotope} & \text {(years)} & \text {Quantity} & \text {1000 Years} \\ ^{238} \mathrm{Pu} & 87.74 & &0.1 \mathrm{g} \end{array}$$

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{2}(x-1)$$.

Solve the exponential equation. $$\left(\frac{3}{4}\right)^{x}=\frac{64}{27}$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=345$$ $$x=\frac{4}{5}$$ $$x=14.8$$ $$x=4.3$$ (Function) $$h(x)=1.9 \log _{10} x$$

Use the graph of \(f\) to describe the transformation that yields the graph of \(g .\) Then sketch the graphs of \(f\) and \(g\) by hand. $$f(x)=\left(\frac{1}{2}\right)^{x}, \quad g(x)=\left(\frac{1}{2}\right)^{-(x+4)}$$

The table shows the yearly sales \(S\) (in millions of dollars) of Whole Foods Market for the years 2006 through 2013. (Source: Whole Foods Market) $$\begin{array}{|l|r|}\hline \text { Year } & \text { Salces } \\\\\hline 2006 & 5,607.4 \\\2007 & 6,591.8 \\\2008 & 7,953.9 \\\2009 & 8,031.6 \\\2010 & 9,005.8 \\\2011 & 10,108.0 \\\2012 & 11,699.0 \\\2013 & 12,917.0 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find an exponential model and a power model for the data and identify the coefficient of determination for each model. Let \(t\) represent the year, with \(t=6\) corresponding to 2006 (b) Use the graphing utility to graph each model with the data. (c) Use the coefficients of determination to determine which model fits the data better.

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 2}(x-2)$$ .

Solve the exponential equation. $$e^{x}=14$$

Solve the equation for \(x.\) $$\log _{7} x=\log _{7} 9$$

Show that the value of \(f(x)\) approaches the value of \(g(x)\) as \(x\) increases without bound (a) graphically and (b) numerically. $$f(x)=1+\left(\frac{0.5}{x}\right)^{x}, g(x)=e^{0.5}$$

The table shows the numbers of single beds \(B\) (in thousands) on North American cruise ships from 2007 through 2012. (Source: Cruise Lines International Association) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Beds, } B \\\\\hline 2007 & 260.0 \\\2008 & 270.7 \\\2009 & 284.8 \\\2010 & 307.7 \\\2011 & 321.2 \\\2012 & 333.7 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model, an exponential model, and a logarithmic model for the data and identify the coefficient of determination for each model. Let \(t\) represent the year, with \(t=7\) corresponding to 2007 (b) Which model is the best fit for the data? Explain. (c) Use the model you chose in part (b) to predict the number of beds in 2017 . Is the number reasonable?

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 3}(x+2)$$.

Solve the equation for \(x.\) $$\log _{5} 5=\log _{5} x$$

Solve the exponential equation. $$e^{x}=66$$

Show that the value of \(f(x)\) approaches the value of \(g(x)\) as \(x\) increases without bound (a) graphically and (b) numerically. $$f(x)=1+\left(\frac{3}{x}\right)^{x}, \quad g(x)=e^{3}$$

The populations \(P\) (in thousands) of Luxembourg for the years 1999 through 2013 are shown in the table, where \(t\) represents the year, with \(t=9\) corresponding to 1999. (Source: European Commission Eurostat) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Population, } P \\\\\hline 1999 & 427.4 \\\2000 & 433.6 \\\2001 & 439.0 \\\2002 & 444.1 \\\2003 & 448.3 \\\2004 & 455.0 \\\2005 & 461.2 \\\2006 & 469.1 \\\2007 & 476.2 \\\2008 & 483.8 \\\2009 & 493.5 \\\2010 & 502.1 \\\2011 & 511.8 \\\2012 & 524.9 \\\2013 & 537.0 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (b) Use the regression feature of the graphing utility to find a power model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (c) Use the regression feature of the graphing utility to find an exponential model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (d) Use the regression feature of the graphing utility to find a logarithmic model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (e) Which model is the best fit for the data? Explain. (f) Use each model to predict the populations of Luxembourg for the years 2014 through 2018. (g) Which model is the best choice for predicting the future population of Luxembourg? Explain. (h) Were your choices of models the same for parts (e) and \((g) ?\) If not, explain why your choices were different.

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 4} x^{2}$$.

Solve the equation for \(x.\) $$\log _{4} 4^{2}=x$$

Solve the exponential equation. $$6\left(10^{x}\right)=216$$