Chapter 6

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.25 & 5.75 \\ \hline 2.25 & 8.75 \\ \hline 3.56 & 12.68 \\ \hline 4.2 & 14.6 \\ \hline 5.65 & 18.95 \\ \hline 6.75 & 22.25 \\ \hline 7.25 & 23.75 \\ \hline 8.6 & 27.8 \\ \hline 9.25 & 29.75 \\ \hline 10.5 & 33.5 \\ \hline \end{array} $$

For the following exercises, use logarithms to solve. \(7 e^{3 n-5}+5=-89\)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right)\)

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(f(x)=\ln (2-x)\)

For the following exercises, rewrite each equation in logarithmic form. \(4^{x}=y\)

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=16.5(1.025)^{\frac{1}{x}}\)

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.28 x}}\) What was the initial population of koi?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). Graph the function.

For the following exercises, use logarithms to solve. \(e^{-3 k}+6=44\)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log \left(\sqrt{x^{3} y^{-4}}\right)\)

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(f(x)=\log \left(x-\frac{3}{7}\right)\)

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

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=11,701(0.97)^{t}\)

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.28 x}}\) How many koi will the pond have after one and a half years?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). What is the initial population of fish?

For the following exercises, use logarithms to solve. \(-5 e^{9 x-8}-8=-62\)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln \left(y \sqrt{\frac{y}{1-y}}\right)\)

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(h(x)=-\log (3 x-4)+3\)

For the following exercises, rewrite each equation in logarithmic form. \(m^{-7}=n\)

For the following exercises, find the formula for an exponential function that passes through the two points given. (0,6) and (3,750)

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.28 x}}\) How many months will it take before there are 20 koi in the pond?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). To the nearest tenth, what is the doubling time for the fish population?

For the following exercises, use logarithms to solve. \(-6 e^{9 x+8}+2=-74\)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log \left(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}\right)\)

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(g(x)=\ln (2 x+6)-5\)

For the following exercises, rewrite each equation in logarithmic form. \(19^{x}=y\)

For the following exercises, find the formula for an exponential function that passes through the two points given. (0,2000) and (2,20)

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.28 x}}\) Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). To the nearest whole number, what will the fish population be after 2 years?

For the following exercises, use logarithms to solve. \(2^{x+1}=5^{2 x-1}\)

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. \(\log \left(2 x^{4}\right)+\log \left(3 x^{5}\right)\)

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(f(x)=\log _{3}(15-5 x)+6\)

For the following exercises, find the formula for an exponential function that passes through the two points given. \(\left(-1, \frac{3}{2}\right)\) and (3,24)

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. Graph the population model to show the population over a span of 10 years.

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). To the nearest tenth, how long will it take for the population to reach \(900 ?\)

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-132=0\)

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. \(\ln \left(6 x^{9}\right)-\ln \left(3 x^{2}\right)\)

For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they exist. If they do not exist, write DNE. \(h(x)=\log _{4}(x-1)+1\)

For the following exercises, rewrite each equation in logarithmic form. \(n^{4}=103\)

For the following exercises, find the formula for an exponential function that passes through the two points given. (-2,6) and (3,1)

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. What was the initial population of wolves transported to the habitat?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). What is the carrying capacity for the fish population? Justify your answer using the graph of \(P\).

For the following exercises, use logarithms to solve. \(7 e^{8 x+8}-5=-95\)

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

For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they exist. If they do not exist, write DNE. \(f(x)=\log (5 x+10)+3\)

For the following exercises, rewrite each equation in logarithmic form. \(\left(\frac{7}{5}\right)^{m}=n\)

For the following exercises, find the formula for an exponential function that passes through the two points given. (3,1) and (5,4)

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 wolves will the habitat have after 3 years?

A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many halflives will have passed before the substance decays to 8.3 grams? What is the total time of decay?

For the following exercises, use logarithms to solve. \(10 e^{8 x+3}+2=8\)