Chapter 6

For the following exercises, use the definition of common and natural logarithms to simplify. \(2 \log (.0001)\)

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Graph the logarithmic equation on the scatter diagram.

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (-3 x)=\ln \left(x^{2}-6 x\right)\)

Use the quotient rule for logarithms to find all \(x\) values such that \(\log _{6}(x+2)-\log _{6}(x-3)=1\). Show the steps for solving.

For the following exercises, use the definition of common and natural logarithms to simplify. \(e^{\ln (1.06)}\)

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. \(y=3742(e)^{0.75 t}\)

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of \(100^{\circ}\) Fahrenheit was taken off the stove to cool in a \(69^{\circ} \mathrm{F}\) room. After fifteen minutes, the internal temperature of the soup was \(95^{\circ} \mathrm{F}\). Use Newton's Law of Cooling to write a formula that models this situation.

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3 m\)

Can the power property of logarithms be derived from the power property of exponents using the equation \(b^{x}=m ?\) If not, explain why. If so, show the derivation.

For the following exercises, use the definition of common and natural logarithms to simplify. \(\ln \left(e^{-5.03}\right)\)

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. \(y=150(e)^{\frac{3.25}{t}}\)

For the following exercises, refer to Table 10 . $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 7.5 & 6 & 5.2 & 4.3 & 3.9 & 3.4 & 3.1 & 2.9 \\ \hline \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (x-2)-\ln (x)=\ln (54)\)

Prove that \(\log _{b}(n)=\frac{1}{\log _{n}(b)}\) for any positive integers \(b>1\) and \(n>1\)

For the following exercises, sketch the graph of the indicated function. \(f(x)=\log _{2}(x+2)\)

For the following exercises, use the definition of common and natural logarithms to simplify. \(e^{\ln (10.125)}+4\)

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. \(y=2.25(e)^{-2 t}\)

For the following exercises, refer to Table 10 . $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 7.5 & 6 & 5.2 & 4.3 & 3.9 & 3.4 & 3.1 & 2.9 \\ \hline \end{array} $$ Use the LOGarithm option of the REGression feature to find a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of \(100^{\circ}\) Fahrenheit was taken off the stove to cool in a \(69^{\circ} \mathrm{F}\) room. After fifteen minutes, the internal temperature of the soup was \(95^{\circ} \mathrm{F}\). To the nearest degree, what will the temperature be after 2 and a half hours?

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{9}\left(2 n^{2}-14 n\right)=\log _{9}\left(-45+n^{2}\right)\)

Does \(\log _{81}(2401)=\log _{3}(7) ?\) Verify the claim algebraically.

For the following exercises, sketch the graph of the indicated function. \(f(x)=2 \log (x)\)

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(\log _{3}\left(\frac{1}{27}\right)\)

Suppose an investment account is opened with an initial deposit of \(\$ 12,000\) earning \(7.2 \%\) interest compounded continuously. How much will the account be worth after 30 years?

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln \left(x^{2}-10\right)+\ln (9)=\ln (10)\)

For the following exercises, sketch the graph of the indicated function. \(f(x)=\ln (-x)\)

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(\log _{6}(\sqrt{6})\)

For the following exercises, evaluate the exponential functions for the indicated value of \(x\). \(g(x)=\frac{1}{3}(7)^{x-2} \quad\) for \(\quad g(6)\).

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of \(165^{\circ} \mathrm{F}\) and is allowed to cool in a \(75^{\circ} \mathrm{F}\) room. After half an hour, the internal temperature of the turkey is \(145^{\circ} \mathrm{F}\). To the nearest degree, what will the temperature be after 50 minutes?

For the following exercises, solve each equation for \(x\). \(\log (x+12)=\log (x)+\log (12)\)

For the following exercises, sketch the graph of the indicated function. \(g(x)=\log (4 x+16)+4\)

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(\log _{2}\left(\frac{1}{8}\right)+4\)

For the following exercises, evaluate the exponential functions for the indicated value of \(x\). \(f(x)=4(2)^{x-1}-2\) for \(f(5)\)

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=2(5)^{x},\) for \(f(-3)\)

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of \(165^{\circ} \mathrm{F}\) and is allowed to cool in a \(75^{\circ} \mathrm{F}\) room. After half an hour, the internal temperature of the turkey is \(145^{\circ} \mathrm{F}\). To the nearest minute, how long will it take the turkey to cool to \(110^{\circ} \mathrm{F} ?\)

For the following exercises, solve each equation for \(x\). \(\ln (x)+\ln (x-3)=\ln (7 x)\)

For the following exercises, sketch the graph of the indicated function. \(g(x)=\log (6-3 x)+1\)

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(6 \log _{8}(4)\)

For the following exercises, evaluate the exponential functions for the indicated value of \(x\). \(h(x)=-\frac{1}{2}\left(\frac{1}{2}\right)^{x}+6\) for $$ h(-7) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-4^{2 x+3},\) for \(f(-1)\)

For the following exercises, refer to \(\underline{\text { Table } 11 .}\) $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8.7 & 12.3 & 15.4 & 18.5 & 20.7 & 22.5 & 23.3 & 24 & 24.6 & 24.8 \\ \hline \end{array} $$ Use a graphing calculator to create a scatter diagram of the data.

For the following exercises, solve each equation for \(x\). \(\log _{2}(7 x+6)=3\)

For the following exercises, sketch the graph of the indicated function. \(h(x)=-\frac{1}{2} \ln (x+1)-3\)

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (10,000)\)

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(-50=-\left(\frac{1}{2}\right)^{-x}\)

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

For the following exercises, refer to \(\underline{\text { Table } 11 .}\) $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8.7 & 12.3 & 15.4 & 18.5 & 20.7 & 22.5 & 23.3 & 24 & 24.6 & 24.8 \\ \hline \end{array} $$ 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, solve each equation for \(x\). \(\ln (7)+\ln \left(2-4 x^{2}\right)=\ln (14)\)

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (0.001)\)