Chapter 4

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln (x+5)=\ln (x-1)-\ln (x+1)\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \frac{z}{\sqrt[3]{z+3}}\)

The population of a town will double in \(t=\frac{8 \ln 3}{\ln 63-\ln 45}\) years. Find \(t\).

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln (x+1)-\ln (x-2)=\ln x$$\ln (x+1)-\ln (x-2)=\ln x\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{9} \frac{\sqrt{y}}{z^{2}}\)

The work \(W\) (in foot-pounds) done in compressing a volume of 9 cubic feet at a pressure of 15 pounds per square inch to a volume of 3 cubic feet is \(W=19,440(\ln 9-\ln 3)\). Find \(W\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{2}(2 x-3)=\log _{2}(x+4)\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt[3]{\frac{x}{y}}\)

Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average score \(g\) for the class can be approximated by the human memory model \(g(t)=78-14 \log _{10}(t+1), \quad 0 \leq t \leq 12\) where \(t\) is the time (in months). (a) What was the average score on the original exam? (b) What was the average score after 4 months? (c) When did the average score drop below 70 ?

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{3}(x+8)=\log _{3}(3 x+2)\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt{\frac{x^{2}}{y^{3}}}\)

Students in a seventh-grade class were given an exam. During the next 2 years, the same students were retested several times. The average score \(g\) can be approximated by the model \(g(t)=87-16 \log _{10}(t+1), \quad 0 \leq t \leq 24\) where \(t\) is the time (in months). (a) What was the average score on the original exam? (b) What was the average score after 6 months? (c) When did the average score drop below \(70 ?\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10}(x+4)-\log _{10} x=\log _{10}(x+2)\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt[4]{x^{3}\left(x^{2}+3\right)}\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} x+\log _{10}(x+1)=\log _{10}(x+3)\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt{x^{2}(x+2)}\)

A principal \(P\), invested at \(4.85 \%\) interest and compounded continuously, increases to an amount that is \(K\) times the principal after \(t\) years, where \(t\) is given by \(t=\frac{\ln K}{0.0485}\) Use a graphing utility to graph this function.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{4} x-\log _{4}(x-1)=\frac{1}{2}\)

In Exercises \(87-102\), condense the expression to the logarithm of a single quantity.\(\log _{3} x+\log _{3} 5\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{3} x+\log _{3}(x-8)=2\)

Condense the expression to the logarithm of a single quantity.\(\log _{5} y+\log _{5} x\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2\)

Condense the expression to the logarithm of a single quantity.\(\log _{4} 8-\log _{4} x\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 4 x-\log _{10}(12+\sqrt{x})=2\)

Condense the expression to the logarithm of a single quantity.\(\log _{10} 4-\log _{10} z\)

Solve for \(y\) in terms of \(x\).\(\ln y=\ln (2 x+1)+\ln 1\)

Condense the expression to the logarithm of a single quantity.\(2 \log _{10}(x+4)\)

Median Age of U.S. Population The model \(A=15.68-0.037 t+6.131 \ln t, \quad 10 \leq t \leq 80\) approximates the median age \(A\) of the United States population from 1980 to \(2050 .\) In the model, \(t\) represents the year, with \(t=10\) corresponding to 1980 (see figure). (Source: U.S. Census Bureau)

Solve for \(y\) in terms of \(x\).\(\ln y=2 \ln x+\ln (x-3)\)

The model \(t=12.542 \ln \left(\frac{x}{x-1000}\right), \quad x>1000\) approximates the length of a home mortgage of \(\$ 150,000\) at \(8 \%\) interest 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) (see figure). (a) Use the model to approximate the length of a \(\$ 150,000\) mortgage at \(8 \%\) interest when the monthly payment is \(\$ 1100.65\) and when the monthly payment is \(\$ 1254.68\). (b) Approximate the total amount paid over the term of the mortgage with a monthly payment of \(\$ 1100.65\) and with a monthly payment of \(\$ 1254.68\). (c) Approximate the total interest charge for a monthly payment of \(\$ 1100.65\) and for a monthly payment of \(\$ 1254.68\) (d) What is the vertical asymptote of the model? Interpret its meaning in the context of the problem.

Solve for \(y\) in terms of \(x\).\(\log _{10} y=2 \log _{10}(x-1)-\log _{10}(x+2)\)

Condense the expression to the logarithm of a single quantity.\(-\ln x-3 \ln 6\)

Solve for \(y\) in terms of \(x\).\(\log _{10}(y-4)+\log _{10} x=3 \log _{10} x\)

Condense the expression to the logarithm of a single quantity.\(2 \ln 8+5 \ln z\)

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(2^{x}-7=0\)

Condense the expression to the logarithm of a single quantity.\(\frac{1}{3} \ln 5 x-\ln (x+1)\)

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(500-1500 e^{-x / 2}=0\)

Condense the expression to the logarithm of a single quantity.\(\frac{3}{2} \ln (z-2)+\ln z$$\frac{3}{2} \ln (z-2)+\ln z\)

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(3-\ln x=0\)

Condense the expression to the logarithm of a single quantity.\(\log _{8}(x-2)-\log _{8}(x+2)\)

Use a graphing utility to solve the equation. Approximate the result to three decimal places. Verify your result algebraically.\(10-4 \ln (x-2)=0\)

Condense the expression to the logarithm of a single quantity.\(3 \log _{7} x+2 \log _{7} y-4 \log _{7} z\)

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.0625\)

Condense the expression to the logarithm of a single quantity.\(2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right)\)

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.085\)

Condense the expression to the logarithm of a single quantity.\(\frac{3}{2} \ln t^{6}-\frac{3}{4} \ln t^{4}\)

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.0725\)

Condense the expression to the logarithm of a single quantity.\(\ln x-\ln (x+2)-\ln (x-2)\)

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.0875\)

Condense the expression to the logarithm of a single quantity.\(\ln (x+1)+2 \ln (x-1)+3 \ln x\)