Chapter 8

In \(53-56,\) find each value of \(x\) to the nearest thousandth. $$ e^{x}=35 $$

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \frac{a^{2}}{10} $$

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \frac{\log _{5} 25+2 \log _{10} 10}{\log _{16} 4} $$

Solve each equation for the variable. \(\log _{2} 8+\log _{3} 9=\log _{b} 100,000\)

In \(53-56,\) find each value of \(x\) to the nearest thousandth. $$ e^{x}=217 $$

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \left(\frac{a}{10}\right)^{2} $$

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \frac{2 \log _{1.5} 2.25}{\log _{4} 64-\log _{80} 80} $$

In \(53-56,\) find each value of \(x\) to the nearest thousandth. $$ e^{x}=2 $$

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \sqrt{a} $$

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \frac{3 \log _{3} 9 \cdot 4 \log _{8} 8 \cdot \log _{13} 169}{6 \log _{2} 256+\log _{\frac{1}{2}} 8} $$

Write the following expression as a single logarithm: \(\log \left(x^{2}-4\right)+2 \log 8-\log 6\)

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \frac{\log _{3} 27+8 \log _{16} 2}{\log _{8} 512} \cdot \log _{1,000} 10 $$

Write the following expression as a single logarithm: \(\frac{1}{2} \ln x-\ln y+\ln z^{3}\)

Write the following expression as a multiple, sum, and/or difference of logarithms: \(\log \sqrt{\frac{x y}{z}}\)

In \(57-68,\) solve each equation for the variable. $$ \log _{10} x=3 $$

Write the following expression as a multiple, sum, and/or difference of logarithms: \(\ln \frac{e^{2} x y^{2}}{z}\)

The formula \(t=\frac{\log K}{0.045 \log e}\) gives the time \(t\) (in years) that it will take an investment \(P\) that is compounded continuously at a rate of 4.5\(\%\) to increase to an amount \(K\) times the original principal. a. Use the formula to complete the table to three decimal places. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline K & {1} & {2} & {3} & {4} & {5} & {10} & {20} & {30} \\ \hline t & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ b. Use the table to graph the function \(t=\frac{\log K}{0.045 \log e}\) c. If Paul invests \(\$ 1,000\) in a savings account that is compounded continuously at a rate of \(4.5 \%,\) when will his investment double? triple?

In \(57-68,\) solve each equation for the variable. $$ \log _{2} 32=x $$

The pH (hydrogen potential) measures the acidity or alkalinity of a solution. In general, acids have pH values less than \(7,\) while alkaline solutions (bases) have pH values greater than \(7 .\) Pure water is considered neutral with a \(\mathrm{pH}\) of \(7 .\) The pH of a solution is given by the formula \(\mathrm{pH}=-\log x\) where \(x\) represents the hydronium ion concentration of the solution. Find, to the nearest hundredth, the approximate \(\mathrm{pH}\) of each of the following: a. Blood: \(x=3.98 \times 10^{-8}\) b. Vinegar: \(x=6.4 \times 10^{-3}\) c. A solution with \(x=4.0 \times 10^{-5}\)

In \(57-68,\) solve each equation for the variable. $$ \log _{5} 625=x $$

In \(57-68,\) solve each equation for the variable. $$ a=\log _{4} 16 $$

In \(57-68,\) solve each equation for the variable. $$ \log _{b} 27=3 $$

In \(57-68,\) solve each equation for the variable. $$ \log _{b} 64=6 $$

In \(57-68,\) solve each equation for the variable. $$ \log _{5} y=-2 $$

In \(57-68,\) solve each equation for the variable. $$ \log _{25} c=-4 $$

In \(57-68,\) solve each equation for the variable. $$ \log _{100} x=-\frac{1}{2} $$

In \(57-68,\) solve each equation for the variable. $$ \log _{8} x=\frac{1}{2} $$

In \(57-68,\) solve each equation for the variable. $$ \log _{36} 6=x $$

If \(\mathrm{f}(x)=\log _{3} x,\) find \(\mathrm{f}(81)\)

If \(\mathrm{p}(x)=\log _{25} x,\) find \(\mathrm{p}(5)\)

If \(\mathrm{g}(x)=\log _{10} x,\) find \(\mathrm{g}(0.001)\)

If \(\mathrm{h}(x)=\log _{32} x,\) find \(\mathrm{h}(8)\)

Solve for \(a : \log _{5} 0.2=\log _{a} 10\)

Solve for \(x : \log _{100} 10=\log _{16} x\)

When \(\$ 1\) is invested at 6\(\%\) interest, its value, \(A,\) after \(t\) years is \(A=1.06^{t} .\) Express \(t\) in terms of \(A .\)

\(R\) is the ratio of the population of a town \(n\) years from now to the population now. If the population has been decreasing by 3\(\%\) each year, \(R=0.97^{n} .\) Express \(n\) in terms of \(R .\)

The decay constant of radium is \(-0.0004\) per year. The amount of radium, \(A,\) present after \(t\) years in a sample that originally contained 1 gram of radium is \(A=e^{-0.0004 t} .\) a. Express \(-0.0004 t\) in terms of \(A\) and \(e .\) b. Solve for \(t\) in terms of \(A\) .