Chapter 6

Solve each equation. Do not use a calculator. $$\left(\frac{1}{2}\right)^{x-6}=8^{x+1} $$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. entering a room

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=4^{x}-3$$

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

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log (x-21)=2$$

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

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

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=\left(\frac{1}{2}\right)^{x}-5$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log (3 x-13)=1$$

Solve each equation. Do not use a calculator. $$(\sqrt[3]{5})^{-x}=\left(\frac{1}{5}\right)^{x+2}$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. wrapping a package

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=-10^{x}+4$$

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

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

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. putting on a coat

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=-e^{x}+6$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (5+4 x)-\ln (3+x)-\ln 3=0$$

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

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

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=3 x-4$$

Suppose \(f(x)=\log _{a} x\) and \(f(3)=2 .\) Determine each function value. $$f\left(\frac{1}{9}\right)$$

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

Solve each equation. Do not use a calculator. $$_{0}^{1-x}=\left(\frac{1}{36}\right)^{2 x} $$

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=4 x-5$$

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

Solve each equation. Do not use a calculator. $$\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5 x}$$

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=x^{3}+1$$

Find the \(p H\) for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Grapefruit, \(6.3 \times 10^{-4}\)

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}(4 x)-\log _{7}(x+3)=\log _{7} x$$

Solve each equation in part (a) analyrically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(2^{x+1}=8\) (b) \(2^{x+1}>8\) (c) \(2^{x+1}<8\)

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=-x^{3}-2$$

Find the \(p H\) for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Limes, \(1.6 \times 10^{-2}\)

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}(2 x)+\log _{2}(x+2)=\log _{2} 16$$

Solve each equation in part (a) analyrically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(3^{2-x}=9\) (b) \(3^{2-x}>9\) (c) \(3^{2-x}<9\)

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$\log x=\frac{1}{2} x-1$$

Find the \(p H\) for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Crackers, \(3.9 \times 10^{-9}\)

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln e^{x}-2 \ln e=\ln e^{4}$$

Solve each equation in part (a) analyrically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(27^{4 x}=9^{x+1}\) (b) \(27^{4 x}>9^{x+1}\) (c) \(27^{4 x}<9^{x+1}\)

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$2^{-x}=\log x$$

Find the \(p H\) for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Sodium hydroxide, \(3.2 \times 10^{-14}\)

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

Solve each equation in part (a) analyrically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(32^{x}=16^{1-x}\) (b) \(32^{x}>16^{1-x}\) (c) \(32^{x}<16^{1-x}\)

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=\frac{1}{x}$$

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$e^{x}=x^{2}$$

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\). Soda pop, 2.7

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

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=\frac{4}{x}$$

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$e^{3 x}=x^{3}+4$$

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\). Wine, 3.4

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (\ln x)=0$$