Chapter 6

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{2}\left(y^{x}\right)\)

For the following exercises, state the domain and range of the function. \(g(x)=\log _{5}(2 x+9)-2\)

The exposure index \(E I\) for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation \(E I=\log _{2}\left(\frac{f^{2}}{t}\right),\) where \(f\) is the "f-stop" setting on the camera, and \(t\) is the exposure time in seconds. Suppose the fstop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

For the following exercises, rewrite each equation in exponential form. \(\log _{16}(y)=x\)

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. \(f(x)=3\left(\frac{1}{2}\right)^{x}\)

For the following exercises, identify whether the statement represents an exponential function. Explain. The height of a projectile at time \(t\) is represented by the function \(h(t)=-4.9 t^{2}+18 t+40\)

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find the carrying capacity.

For the following exercises, use like bases to solve the exponential equation. \(\frac{36^{3 b}}{36^{2 b}}=216^{2-b}\)

For the following exercises, condense to a single logarithm if possible. \(\ln (7)+\ln (x)+\ln (y)\)

For the following exercises, state the domain and range of the function. \(h(x)=\ln (4 x+17)-5\)

For the following exercises, rewrite each equation in exponential form. \(\log _{x}(64)=y\)

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. \(g(x)=-2(0.25)^{x}\)

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Graph the model.

For the following exercises, use like bases to solve the exponential equation. \(\left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{6}\)

For the following exercises, condense to a single logarithm if possible. \(\log _{3}(2)+\log _{3}(a)+\log _{3}(11)+\log _{3}(b)\)

For the following exercises, state the domain and range of the function. \(f(x)=\log _{2}(12-3 x)-3\)

For the following exercises, rewrite each equation in exponential form. \(\log _{y}(x)=-11\)

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. \(h(x)=6(1.75)^{-x}\)

To the nearest whole number, what is the initial value of a population modeled by the logistic equation \(P(t)=\frac{175}{1+6.995 e^{-0.68 t}} ?\) What is the carrying capacity?

For the following exercises, use logarithms to solve. \(9^{x-10}=1\)

For the following exercises, condense to a single logarithm if possible. \(\log _{b}(28)-\log _{b}(7)\)

For the following exercises, state the domain and the vertical asymptote of the function. \(f(x)=\log _{b}(x-5)\)

For the following exercises, rewrite each equation in exponential form. \(\log _{15}(a)=b\)

For the following exercises, graph each set of functions on the same axes. \(f(x)=3\left(\frac{1}{4}\right)^{x}, g(x)=3(2)^{x},\) and \(h(x)=3(4)^{x}\)

Rewrite the exponential model \(A(t)=1550(1.085)^{x}\) as an equivalent model with base \(e\). Express the exponent to four significant digits.

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Rewrite \(f(x)=1.68(0.65)^{x}\) as an exponential equation with base \(e\) to five significant digits.

For the following exercises, use logarithms to solve. \(2 e^{6 x}=13\)

For the following exercises, condense to a single logarithm if possible. \(\ln (a)-\ln (d)-\ln (c)\)

For the following exercises, state the domain and the vertical asymptote of the function. \(g(x)=\ln (3-x)\)

For the following exercises, rewrite each equation in exponential form. \(\log _{y}(137)=x\)

For the following exercises, graph each set of functions on the same axes. \(f(x)=\frac{1}{4}(3)^{x}, g(x)=2(3)^{x},\) and \(\quad h(x)=4(3)^{x}\)

A logarithmic model is given by the equation \(h(p)=67.682-5.792 \ln (p) .\) To the nearest hundredth, for what value of \(p\) does \(h(p)=62 ?\)

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. $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 2 \\ \hline 2 & 4.079 \\ \hline 3 & 5.296 \\ \hline 4 & 6.159 \\ \hline 5 & 6.828 \\ \hline 6 & 7.375 \\ \hline 7 & 7.838 \\ \hline 8 & 8.238 \\ \hline 9 & 8.592 \\ \hline 10 & 8.908 \\ \hline \end{array} $$

For the following exercises, condense to a single logarithm if possible. \(-\log _{b}\left(\frac{1}{7}\right)\)

For the following exercises, state the domain and the vertical asymptote of the function. \(f(x)=\log (3 x+1)\)

For the following exercises, rewrite each equation in exponential form. \(\log _{13}(142)=a\)

A logistic model is given by the equation \(P(t)=\frac{90}{1+5 e^{-0.42 t}} .\) To the nearest hundredth, for what value of \(t\) does \(P(t)=45 ?\)

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. $$ \begin{array}{c|c} \hline x & f(x) \\ \hline 1 & 2.4 \\ \hline 2 & 2.88 \\ \hline 3 & 3.456 \\ \hline 4 & 4.147 \\ \hline 5 & 4.977 \\ \hline 6 & 5.972 \\ \hline 7 & 7.166 \\ \hline 8 & 8.6 \\ \hline 9 & 10.32 \\ \hline 10 & 12.383 \\ \hline \end{array} $$

For the following exercises, use logarithms to solve. \(2 \cdot 10^{9 a}=29\)

For the following exercises, condense to a single logarithm if possible. \(\frac{1}{3} \ln (8)\)

For the following exercises, state the domain and the vertical asymptote of the function. \(f(x)=3 \log (-x)+2\)

For the following exercises, rewrite each equation in exponential form. \(\log (v)=t\)

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=300(1-t)^{5}\)

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. $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 4 & 9.429 \\ \hline 5 & 9.972 \\ \hline 6 & 10.415 \\ \hline 7 & 10.79 \\ \hline 8 & 11.115 \\ \hline 9 & 11.401 \\ \hline 10 & 11.657 \\ \hline 11 & 11.889 \\ \hline 12 & 12.101 \\ \hline 13 & 12.295 \\ \hline \end{array} $$

For the following exercises, use logarithms to solve. \(-8 \cdot 10^{p+7}-7=-24\)

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. \(\log \left(\frac{x^{15} y^{13}}{z^{19}}\right)\)

For the following exercises, state the domain and the vertical asymptote of the function. \(g(x)=-\ln (3 x+9)-7\)

For the following exercises, rewrite each equation in exponential form. \(\ln (w)=n\)

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=220(1.06)^{x}\)

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.28 x}}\) Graph the population model to show the population over a span of 3 years.