Chapter 6

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(116=\frac{1}{4}\left(\frac{1}{8}\right)^{x}\)

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-2 e^{x-1},\) for \(f(-1)\)

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: \(10^{-10} \frac{W}{m^{2}},\) Vacuum: \(10^{-4} \frac{W}{m^{2}},\) Jet: \(10^{2} \frac{W}{m^{2}}\)

For the following exercises, solve each equation for \(x\). \(\log _{8}(x+6)-\log _{8}(x)=\log _{8}(58)\)

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (1)+7\)

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(12=2(3)^{x}+1\)

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=2.7(4)^{-x+1}+1.5,\) for \(f(-2)\)

For the following exercises, refer to \(\underline{\text { Table } 11 .}\) $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8.7 & 12.3 & 15.4 & 18.5 & 20.7 & 22.5 & 23.3 & 24 & 24.6 & 24.8 \\ \hline \end{array} $$ To the nearest whole number, what is the predicted carrying capacity of the model?

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: \(10^{-10} \frac{W}{m^{2}},\) Vacuum: \(10^{-4} \frac{W}{m^{2}},\) Jet: \(10^{2} \frac{W}{m^{2}}\)

For the following exercises, solve each equation for \(x\). \(\ln (3)-\ln (3-3 x)=\ln (4)\)

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(2 \log \left(100^{-3}\right)\)

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(5=3\left(\frac{1}{2}\right)^{x-1}-2\)

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=1.2 e^{2 x}-0.3,\) for \(f(3)\)

For the following exercises, use this scenario: The equation \(N(t)=\frac{500}{1+49 e^{-0.7 t}}\) models the number of people in a town who have heard a rumor after t days. How many people started the rumor?

For the following exercises, solve each equation for \(x\). \(\log _{3}(3 x)-\log _{3}(6)=\log _{3}(77)\)

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(\ln \left(e^{\frac{1}{3}}\right)\)

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(-30=-4(2)^{x+2}+2\)

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-\frac{3}{2}(3)^{-x}+\frac{3}{2},\) for \(f(2)\)

For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.

For the following exercises, use this scenario: The equation \(N(t)=\frac{500}{1+49 e^{-0.7 t}}\) models the number of people in a town who have heard a rumor after t days. To the nearest whole number, how many people will have heard the rumor after 3 days?

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(x)-5=-4\)

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(\log (x-1)+2=\ln (x-1)+2\)

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(\ln (1)\)

Explore and discuss the graphs of \(\quad F(x)=(b)^{x}\) and \(G(x)=\left(\frac{1}{b}\right)^{x}\). Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b}\right)^{x}\) for any real number \(\quad b>0\).

For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ Use the LOGISTIC regression option to find a logistic growth model of the form \(y=\frac{c}{1+a e^{-b x}}\) that best fits the data in the table.

For the following exercises, use this scenario: The equation \(N(t)=\frac{500}{1+49 e^{-0.7 t}}\) models the number of people in a town who have heard a rumor after t days. As \(t\) increases without bound, what value does \(N(t)\) approach? Interpret your answer.

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{3}(x)+3=2\)

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(\log (2 x-3)+2=-\log (2 x-3)+5\)

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(\ln \left(e^{-0.225}\right)-3\)

For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ Graph the logistic equation on the scatter diagram.

A doctor injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient's system. Which is an appropriate model for this situation? (a) \(f(t)=13(0.0805)^{t}\) (b) \(f(t)=13 e^{0.9195 t}\) (c) \(f(t)=13 e^{(-0.0839 t)}\) (d) \(f(t)=\frac{4.75}{1+13 e^{-0.83925 t}}\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\ln (3 x)=2\)

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(\ln (x-2)=-\ln (x+1)\)

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(25 \ln \left(e^{\frac{2}{5}}\right)\)

Explore and discuss the graphs of \(f(x)=4^{x},\) \(g(x)=4^{x-2},\) and \(\quad h(x)=\left(\frac{1}{16}\right) 4^{x} .\) Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b^{n}}\right) b^{x} \quad\) for any real number \(n\) and real number \(\quad b>0\).

For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ To the nearest whole number, what is the predicted carrying capacity of the model?

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\ln (x-5)=1\)

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(2 \ln (5 x+1)=\frac{1}{2} \ln (-5 x)+1\)

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\log (0.04)\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log (4)+\log (-5 x)=2\)

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(\frac{1}{3} \log (1-x)=\log (x+1)+\frac{1}{3}\)

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\ln (15)\)

Recall that the general form of a logistic equation for a population is given by \(P(t)=\frac{c}{1+a e^{-b t}},\) such that the initial population at time \(t=0\) is \(P(0)=P_{0} .\) Show algebraically that \(\frac{c-P(t)}{P(t)}=\frac{c-P_{0}}{P_{0}} e^{-b t}\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(-7+\log _{3}(4-x)=-6\)

Let \(b\) be any positive real number such that \(b \neq 1\). What must \(\log _{b} 1\) be equal to? Verify the result.

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\ln \left(\frac{4}{5}\right)\)

The annual percentage yield (APY) of an investment account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula \(\mathrm{APY}=\left(1+\frac{r}{12}\right)^{12}-1\)

Use a graphing utility to find an exponential regression formula \(f(x)\) and a logarithmic regression formula \(g(x)\) for the points (1.5,1.5) and \((8.5,8.5) .\) Round all numbers to 6 decimal places. Graph the points and both formulas along with the line \(y=x\) on the same axis. Make a conjecture about the relationship of the regression formulas.

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\ln (4 x-10)-6=-5\)

Explore and discuss the graphs of \(f(x)=\log _{\frac{1}{2}}(x)\) and \(g(x)=-\log _{2}(x) .\) Make a conjecture based on the result.