91Ó°ÊÓ

Solve the equation. Write the solution set with exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(8^{y^{2}-7}=64\)

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{4}(3 d)+\log _{4} 1=\log _{4}(3 d) $$

Scientists use the pH scale to represent the level of acidity or alkalinity of a liquid. This is based on the molar concentration of hydrogen ions, \(\left[\mathrm{H}^{+}\right] .\) Since the values of \(\left[\mathrm{H}^{+}\right]\) vary over a large range, \(1 \times 10^{0}\) mole per liter to \(1 \times 10^{-14}\) mole per liter (mol/L), a logarithmic scale is used to compute \(\mathrm{pH}\). The formula $$\mathbf{p} \mathbf{H}=-\log \left[\mathbf{H}^{+}\right]$$ represents the pH of a liquid as a function of its concentration of hydrogen ions, \(\left[\mathrm{H}^{+}\right]\) The \(\mathrm{pH}\) scale ranges from 0 to \(14 .\) Pure water is taken as neutral having a pH of 7 . A pH less than 7 is acidic. A pH greater than 7 is alkaline (or basic). For Exercises \(97-98,\) use the formula for \(\mathrm{pH}\). Round \(\mathrm{pH}\) values to 1 decimal place. Bleach and milk of magnesia are both bases. Their \(\left[\mathrm{H}^{+}\right]\) values are \(2.0 \times 10^{-13} \mathrm{~mol} / \mathrm{L}\) and \(4.1 \times 10^{-10}\) mol/L, respectively. a. Find the \(\mathrm{pH}\) for bleach. b. Find the \(\mathrm{pH}\) for milk of magnesia. c. Which substance is more basic?

Show that every increasing function is one-to-one.

Solve the equation. Write the solution set with exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{9}|x+4|=\log _{9} 6\)

Explain why the product property of logarithms does not apply to the following statement. $$ \begin{array}{l} \log _{5}(-5)+\log _{5}(-25) \\ \quad=\log _{5}[(-5)(-25)] \\ \quad=\log _{5} 125=3 \end{array} $$

A function is said to be periodic if there exists some nonzero real number \(p,\) called the period, such that \(f(x+p)=f(x)\) for all real numbers \(x\) in the domain of \(f\). Explain why no periodic function is one-to-one.

Explain how to use the change-of-base formula and explain why it is important.

Solve the equation. Write the solution set with exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{8}|3-x|=\log _{8} 5\)

a. Write the difference quotient for \(f(x)=\ln x\). b. Show that the difference quotient from part (a) can be written as \(\ln \left(\frac{x+h}{x}\right)^{1 / h}\).