91Ó°ÊÓ

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

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(\left(\frac{1}{2}\right)^{-x}=\left(\frac{1}{4}\right)^{x+1}\) (b) \(\left(\frac{1}{2}\right)^{-x} \geq\left(\frac{1}{4}\right)^{x+1}\) (c) \(\left(\frac{1}{2}\right)^{-x} \leq\left(\frac{1}{4}\right)^{x+1}\)

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 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. $$y=\frac{1}{x}$$

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

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 equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(\left(\frac{2}{3}\right)^{x-1}=\left(\frac{81}{16}\right)^{x+1}\) (b) \(\left(\frac{2}{3}\right)^{x-1} \leq\left(\frac{81}{16}\right)^{x+1}\) (c) \(\left(\frac{2}{3}\right)^{x-1} \geq\left(\frac{81}{16}\right)^{x+1}\)

For each function that is one-to-one, write an equation for the inverse function 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. \(\log _{10} x=x-2\)

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