Chapter 6

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. \(\log (x)-\frac{1}{2} \log (y)+3 \log (z)\)

For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they exist. If they do not exist, write DNE. \(g(x)=\ln (-x)-2\)

For the following exercises, rewrite each equation in logarithmic form. \(y^{x}=\frac{39}{100}\)

For the following exercises, graph the function and its reflection about the \(x\) -axis on the same axes. \(f(x)=\frac{1}{2}(4)^{x}\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline f(x) & 70 & 40 & 10 & -20 \\ \hline \end{array} $$

For the following exercises, use this scenario: The population \(P\) of an endangered species habitat for wolves is modeled by the function \(P(x)=\frac{558}{1+54.8 e^{-0.462 x}},\) where \(x\) is given in years. How many years will it take before there are 100 wolves in the habitat?

The formula for an increasing population is given by \(P(t)=P_{0} e^{r t}\) where \(P_{0}\) is the initial population and \(r>0\). Derive a general formula for the time \(t\) it takes for the population to increase by a factor of \(M\).

For the following exercises, use logarithms to solve. \(4 e^{3 x+3}-7=53\)

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. \(4 \log _{7}(c)+\frac{\log _{7}(a)}{3}+\frac{\log _{7}(b)}{3}\)

For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they exist. If they do not exist, write DNE. \(f(x)=\log _{2}(x+2)-5\)

For the following exercises, rewrite each equation in logarithmic form. \(10^{a}=b\)

For the following exercises, graph the function and its reflection about the \(x\) -axis on the same axes. \(f(x)=3(0.75)^{x}-1\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ h(x) & 70 & 49 & 34.3 & 24.01 \end{array} $$

Recall the formula for calculating the magnitude of an earthquake, \(M=\frac{2}{3} \log \left(\frac{S}{S_{0}}\right)\). Show each step for solving this equation algebraically for the seismic moment \(S\).

For the following exercises, use logarithms to solve. \(8 e^{-5 x-2}-4=-90\)

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. \(\log _{7}(15)\) to base \(e\)

For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they exist. If they do not exist, write DNE. \(h(x)=3 \ln (x)-9\)

For the following exercises, rewrite each equation in logarithmic form. \(e^{k}=h\)

For the following exercises, graph the function and its reflection about the \(x\) -axis on the same axes. \(f(x)=-4(2)^{x}+2\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ \hline m(x) & 80 & 61 & 42.9 & 25.61 \end{array} $$

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.

What is the \(y\) -intercept of the logistic growth model \(y=\frac{c}{1+a e^{-r x}} ?\) Show the steps for calculation. What does this point tell us about the population?

For the following exercises, use logarithms to solve. \(3^{2 x+1}=7^{x-2}\)

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. \(\log _{14}(55.875)\) to base 10

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{3}(x)=2\)

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. \(f(x)=2^{-x}\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline f(x) & 10 & 20 & 40 & 80 \\ \hline \end{array} $$

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Use the regression feature to find an exponential function that best fits the data in the table.

Prove that \(b^{x}=e^{x \ln (b)}\) for positive \(b \neq 1\).

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-6=0\)

For the following exercises, suppose \(\log _{5}(6)=a\) and \(\log _{5}(11)=b .\) Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b .\) Show the steps for solving. \(\log _{11}(5)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{2}(x)=-3\)

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. \(h(x)=2^{x}+3\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{cc|c|cc} x & 1 & 2 & 3 & 4 \\ \hline g(x) & -3.25 & 2 & 7.25 & 12.5 \end{array} $$

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Write the exponential function as an exponential equation with base \(e\).

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about \(30 \%\) each hour. To the nearest hour, what is the half-life of the drug?

For the following exercises, suppose \(\log _{5}(6)=a\) and \(\log _{5}(11)=b .\) Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b .\) Show the steps for solving. \(\log _{6}(55)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{5}(x)=2\)

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. \(f(x)=2^{x-2}\)

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about \(30 \%\) each hour. Write an exponential model representing the amount of the drug remaining in the patient's system after \(t \quad\) hours. Then use the formula to find the amount of the drug that would remain in the patient's system after 3 hours. Round to the nearest milligram.

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\)

For the following exercises, suppose \(\log _{5}(6)=a\) and \(\log _{5}(11)=b .\) Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b .\) Show the steps for solving. \(\log _{11}\left(\frac{6}{11}\right)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{3}(x)=3\)

For the following exercises, describe the end behavior of the graphs of the functions. \(f(x)=-5(4)^{x}-1\)

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Use the intersect feature to find the value of \(x\) for which \(f(x)=4000\)

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log _{324}(18)=\frac{1}{2}\)

For the following exercises, use properties of logarithms to evaluate without using a calculator. \(\log _{3}\left(\frac{1}{9}\right)-3 \log _{3}(3)\)

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{2}(x)=6\)

For the following exercises, describe the end behavior of the graphs of the functions. \(f(x)=3\left(\frac{1}{2}\right)^{x}-2\)

For the following exercises, refer to Table \(8 .\) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.