Chapter 4

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10}\left(z^{2}+19\right)=2$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\ln \frac{x+2}{x-1}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the expressions equivalent? Explain. Verify your conclusion algebraically. $$y_{1}=\ln x^{2}, \quad y_{2}=2 \ln x$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{12} x^{2}=6$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\ln \frac{2 x}{x+2}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the expressions equivalent? Explain. Verify your conclusion algebraically. $$y_{1}=2(\ln 2+\ln x), \quad y_{2}=\ln 4 x^{2}$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln \sqrt{x+2}=1$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\ln \frac{x^{2}}{10}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the expressions equivalent? Explain. Verify your conclusion algebraically. $$y_{1}=\ln (x-2)+\ln (x+2), \quad y_{2}=\ln \left(x^{2}-4\right)$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln \sqrt{x-8}=5$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\ln \frac{x}{x^{2}+1}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the expressions equivalent? Explain. Verify your conclusion algebraically. $$y_{1}=\frac{1}{4} \ln \left[x^{4}\left(x^{2}+1\right)\right], \quad y_{2}=\ln x+\frac{1}{4} \ln \left(x^{2}+1\right)$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln (x+1)^{2}=2$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\sqrt{\ln x}$$

The relationship between the number of decibels \(\beta\) and the intensity of a sound \(I\) in watts per square meter is given by $$\beta=10 \log _{10}\left(\frac{I}{10^{-12}}\right)$$.(a) Use the properties of logarithms to write the formula in a simpler form. (b) Use a graphing utility to complete the table. Verify your answers algebraically.$$\begin{array}{|l|l|l|l|l|l|l|}\hline I & 10^{-4} & 10^{-6} & 10^{-8} & 10^{-10} & 10^{-12} & 10^{-14} \\\\\hline \beta & & & & & & \\\\\hline\end{array}$$.

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=(\ln x)^{2}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

A beaker of liquid at an initial temperature of \(78^{\circ} \mathrm{C}\) is placed in a room at a constant temperature of \(21^{\circ} \mathrm{C}\). The temperature of the liquid is measured every 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form \((t, T),\) where \(t\) is the time (in minutes) and \(T\) is the temperature (in degrees Celsius). (Spreadsheet atLarsonPrecalculus.com).\(\left(0,78.0^{\circ}\right),\left(5,66.0^{\circ}\right),\left(10,57.5^{\circ}\right),\left(15,51.2^{\circ}\right)\) ,\(\left(20,46.3^{\circ}\right),\left(25,42.5^{\circ}\right),\left(30,39.6^{\circ}\right)\). (a) The graph of the temperature of the room should be an asymptote of the graph of the model for the data. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points \((t, T)\) and \((t, T-21)\). (b) An exponential model for the data \((t, T-21)\) is given by $$T-21=54.4(0.964)^{t}$$.Solve for \(T\) and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use the graphing utility to plot the points \((t, \ln (T-21))\) and observe that the points appear linear. Use the regression feature of the graphing utility to fit a line to the data. The resulting line has the form $$\ln (T-21)=a t+b$$.Use the properties of logarithms to solve for \(T\) Verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the \(y\) -coordinates of the revised data points to generate the points $$\left(t, \frac{1}{T-21}\right)$$ Use the graphing utility to plot these points and observe that they appear linear. Use the regression feature of the graphing utility to fit a line to the data. The resulting line has the form $$\frac{1}{T-21}=a t+b$$.Solve for \(T\) and use the graphing utility to graph the rational function and the original data points.

Students in a mathematics class were given an exam and then tested monthly with an equivalent exam. The average scores for the class are given by the human memory model $$f(t)=80-17 \log _{10}(t+1), \quad 0 \leq t \leq 12$$ where \(t\) is the time in months. (a) What was the average score on the original exam \((t=0) ?\) (b) What was the average score after 2 months? (c) What was the average score after 11 months? Verify your answers in parts (a), (b), and (c) using a graphing utility.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{3} x+\log _{3}(x-8)=2$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln (x+5)=\ln (x-1)+\ln (x+1)$$

Determine whether the statement is true or false given that \(f(x)=\ln x\) where \(x>0 .\) Justify your answer.$$f(a x)=f(a)+f(x), a>0$$.

A principal \(P,\) invested at \(3 \frac{1}{2} \%\) and compounded continuously, increases to an amount \(K\) times the original principal after \(t\) years, where \(T=(\ln K) / 0.035.\) (a) Complete the table and interpret your results. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline K & 1 & 2 & 4 & 6 & 8 & 10 & 12 \\\\\hline t & & & & & & & \\\\\hline\end{array}$$ (b) Use a graphing utility to graph the function.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln (x+1)-\ln (x-2)=\ln x$$

The relationship between the number of decibels \(\beta\) and the intensity of a sound \(I\) in watts per square meter is given by $$\beta=10 \log _{10}\left(\frac{I}{10^{-12}}\right).$$ (a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter. (b) Determine the number of decibels of a sound with an intensity of \(10^{-2}\) watt per square meter. (c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2$$

The model $$t=16.625 \ln \frac{x}{x-750}, \quad x>750$$ approximates the length of a home mortgage of \(\$ 150,000\) at \(6 \%\) in terms of the monthly payment. In the model, \(t\) is the length of the mortgage in years and \(x\) is the monthly payment in dollars. (a) Use the model to approximate the lengths of a \(\$ 150,000\) mortgage at \(6 \%\) when the monthly payment is \(\$ 897.72\) and when the monthly payment is \(\$ 1659.24\) (b) Approximate the total amounts paid over the term of the mortgage with a monthly payment of \(\$ 897.72\) and with a monthly payment of \(\$ 1659.24\) What amount of the total is interest costs for each payment?

Determine whether the statement is true or false given that \(f(x)=\ln x\) where \(x>0 .\) Justify your answer.$$\sqrt{f(x)}=\frac{1}{2} f(x)$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 4 x-\log _{10}(12+\sqrt{x})=2$$

The rate of ventilation required in a public school classroom depends on the volume of air space per child. The model $$y=80.4-11 \ln x, \quad 100 \leq x \leq 1500$$ approximates the minimum required rate of ventilation \(y\) (in cubic feet per minute per child) in a classroom with \(x\) cubic feet of air space per child. (a) Use a graphing utility to graph the function and approximate the required rate of ventilation in a room with 300 cubic feet of air space per child. (b) A classroom of 30 students has an air conditioning system that moves 450 cubic feet of air per minute. Determine the rate of ventilation per child. (c) Use the graph in part (a) to estimate the minimum required air space per child for the classroom in part (b). (d) The classroom in part (b) has 960 square feet of floor space and a ceiling that is 12 feet high. Is the rate of ventilation for this classroom adequate? Explain.

Determine whether the statement is true or false given that \(f(x)=\ln x\) where \(x>0 .\) Justify your answer.$$[f(x)]^{n}=n f(x)$$.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$\ln 2 x=2.4$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 3 & 4 & 5 & 6 \\\\\hline \ln 2 x & & & & & \\\\\hline\end{array}$$

Determine whether the statement is true or false. Justify your answer. You can determine the graph of \(f(x)=\log _{6} x\) by graphing \(g(x)=6^{x}\) and reflecting it about the \(x\) -axis.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$3 \ln 5 x=10$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 4 & 5 & 6 & 7 & 8 \\\\\hline 3 \ln 5 x & & & & & \\\\\hline\end{array}$$

Determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{3} x\) contains the point (27,3).

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$6 \log _{3}(0.5 x)=11$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 12 & 13 & 14 & 15 & 16 \\\\\hline 6 \log _{3}(0.5 x) & & & & & \\\\\hline\end{array}$$

Find the value of the base \(b\) so that the graph of \(f(x)=\log _{b} x\) contains the given point. $$(32,5)$$

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$5 \log _{10}(x-2)=11$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 150 & 155 & 160 & 165 & 170 \\\\\hline 5 \log _{10}(x-2) & & & & & \\\\\hline\end{array}$$

Find the value of the base \(b\) so that the graph of \(f(x)=\log _{b} x\) contains the given point. $$(81,4)$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\log _{10} x=x^{3}-3$$

Find the value of the base \(b\) so that the graph of \(f(x)=\log _{b} x\) contains the given point. $$\left(\frac{1}{81}, 2\right)$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\log _{10} x=(x-3)^{2}$$

Find the value of the base \(b\) so that the graph of \(f(x)=\log _{b} x\) contains the given point. $$\left(\frac{1}{64}, 3\right)$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\log _{10} x+e^{0.5 x}=6$$

Think About It Does \(y_{1}=\ln [x(x-2)]\) have the same domain as \(y_{2}=\ln x+\ln (x-2) ?\) Explain.

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$e^{x} \log _{10} x=7$$

Prove that \(\frac{\log _{a} x}{\log _{a / b} x}=1+\log _{a} \frac{1}{b}\).

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln (x+2)-3^{x-2}+10=5$$

Explain why \(\log _{a} x\) is defined only for \(01\).

Simplify the expression.$$\left(64 x^{3} y^{4}\right)^{-3}\left(8 x^{3} y^{2}\right)^{4}$$