Chapter 6

Sketch the graph of \(f(x)=\left(\frac{1}{3}\right)^{x}\). Then refer to it and use earlier techniques to graph each finction. $$f(x)=\left(\frac{1}{3}\right)^{x}+4$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}+4, \quad g(x)=\sqrt[3]{x-4}$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=\log _{2}(x+4)$$

Evaluate each expression. Do not use a calculator. $$\ln e^{2 / 3}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{6}(2 x+4)=2$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}-7, \quad g(x)=\sqrt[3]{x+7}$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=\log _{2}(x-6)$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{5}(8-3 x)=3$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=-x^{5}, \quad g(x)=-\sqrt[5]{x}$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=3 \log _{2} x+1$$

Evaluate each expression. Do not use a calculator. $$\ln e^{x}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{4}\left(x^{3}+37\right)=3$$

Solve each equation. Do not use a calculator. $$2^{3-x}=8$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=-x^{7}, \quad g(x)=-\sqrt[7]{x}$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=-4 \log _{2} x-8$$

Evaluate each expression. Do not use a calculator. $$\ln e^{\sqrt{6}}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}\left(x^{3}+65\right)=0$$

Solve each equation. Do not use a calculator. $$5^{2 x+1}=2$$

Determine whether each finction is one-to-one. If so, find its inverse. $$f=\\{(10,4),(20,5),(30,6),(40,7)\\}$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=\log _{2}(-x)+1$$

Evaluate each expression. Do not use a calculator. $$\sqrt{7} \ln e^{\sqrt{7}}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln x+\ln x^{2}=3$$

Solve each equation. Do not use a calculator. $$12^{x-3}=1$$

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=-\log _{2}(-x)$$

Evaluate each expression. Do not use a calculator. $$\sqrt{2} \ln e^{\sqrt{2}}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log x^{2}=3$$

Solve each equation. Do not use a calculator. $$3^{5-x}=1$$

Graph \(y=\log x^{3}\) and \(y=3 \log x\) on separate sets of axes. It would seem, at first glance, that by applying the power rule for logarithms, these graphs should be the same. Are they? If not, why not? (Hint: Consider the domain in each case.)

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 43$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$2 \ln (x-1)+30=34$$

Solve each equation. Do not use a calculator. $$e^{4 x-1}=\left(e^{2}\right)^{x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 1247$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$1-4 \ln (2 x-1)=-5$$

Solve each equation. Do not use a calculator. $$e^{3-x}=\left(e^{3}\right)^{-x}$$

Evaluate each logarithm in three ways: (a) Use the definition of logarithm to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator. (c) Use a buill-in calculator finction to evaluate the logarithm. (d) Support the result of part (a) by locating the appropriate point on the graph of the finction \(y=\log _{a} x\). $$\log _{9} 27$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 0.783$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$5 \log \left(x^{2}-1\right)+7=12$$

Solve each equation. Do not use a calculator. $$27^{4 x}=9^{x+1}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 0.014$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$8 \log \left(4-x^{2}\right)-4=20$$

Solve each equation. Do not use a calculator. $$32^{x}=16^{1-x}$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. tying your shoelaces

Evaluate each logarithm in three ways: (a) Use the definition of logarithm to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator. (c) Use a buill-in calculator finction to evaluate the logarithm. (d) Support the result of part (a) by locating the appropriate point on the graph of the finction \(y=\log _{a} x\). $$\log _{16}\left(\frac{1}{8}\right)$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 28^{3}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$3 \log _{2}\left(3 x^{2}+2\right)+1=2$$

Solve each equation. Do not use a calculator. $$\left(\frac{1}{4}\right)^{2-x}=2^{3 x+3}$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. pressing a car's accelerator

Evaluate each logarithm in three ways: (a) Use the definition of logarithm to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator. (c) Use a buill-in calculator finction to evaluate the logarithm. (d) Support the result of part (a) by locating the appropriate point on the graph of the finction \(y=\log _{a} x\). $$\log _{2} \sqrt{8}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log (47 \times 93)$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$2 \log _{2}(5 x-3)+1=17$$