Chapter 4

Population Growth The population \(P\) of a small city increases according to the model \(P(t)=36,000 e^{0.0156 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Use the model to predict the population in each year. (a) 2009 (b) 2011 (c) 2015 (d) 2018

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 2 x=7\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{3} 4 n\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(h(x)=\log _{2}(x+4)\)

Radioactive Decay Strontium-90 has a half-life of \(29.1\) years. The amount \(S\) of 100 kilograms of strontium-90 present after \(t\) years is given by \(S=100 e^{-0.0238 t}\) How much of the 100 kilograms will remain after 50 years?

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 3 z=2\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{6} 6 x\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=\log _{4}(x-3)\)

Radioactive Decay Neptunium-237 has a half-life of 2.1 million years. The amount \(N\) of 200 kilograms of neptunium- 237 present after \(t\) years is given by \(N=200 e^{-0.00000033007 t}\) How much of the 200 kilograms will remain after 20,000 years?

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(5 \log _{3}(x+1)=12\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{5} \frac{x}{25}\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=-\log _{2} x\)

Radioactive Decay Five pounds of the element plutonium \(\left({ }^{230} \mathrm{Pu}\right)\) is released in a nuclear accident. The amount of plutonium \(P\) that is present after \(t\) months is given by \(P=5 e^{-0.1507 t}\). (a) Use a graphing utility to graph this function over the interval from \(t=0\) to \(t=10\). (b) How much of the 5 pounds of plutonium will remain after 10 months? (c) Use the graph to estimate the half-life of \({ }^{230} \mathrm{Pu}\). Explain your reasoning.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(5 \log _{10}(x-2)=11\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{10} \frac{y}{2}\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(h(x)=-\log _{4}(x-1)\)

Radioactive Decay One hundred grams of radium \(\left({ }^{226} \mathrm{Ra}\right)\) is stored in a container. The amount of radium \(R\) present after \(t\) years is given by \(R=100 e^{-0.0004335 t}\). (a) Use a graphing utility to graph this function over the interval from \(t=0\) to \(t=10,000\). (b) How much of the 100 grams of radium will remain after 10,000 years? (c) Use the graph to estimate the half-life of \({ }^{226} \mathrm{Ra}\). Explain your reasoning.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(3 \ln 5 x=10\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{2} x^{4}\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(g(x)=\ln (-x)\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(2 \ln x=7\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{2} z^{-3}\)

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=\ln (3-x)\)

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

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

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(h(x)=\ln (x+1)\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln \sqrt{x-8}=5\)

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

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=3+\ln x\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(7+3 \ln x=5\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln x y z\)

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\log (x+1)\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(2-6 \ln x=10\)

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

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\log (x-1)\)

Hospital Employment The numbers of people \(E\) (in thousands) employed in hospitals from 1999 to 2005 can be modeled by \(E=3331(1.0182)^{t}, \quad 9 \leq t \leq 15\) where \(t\) represents the year, with \(t=9\) corresponding to 1999\. (Source: U.S. Bureau of Labor Statistics) (a) Use a graphing utility to graph \(E\) for the years 1999 to \(2005 .\) (b) Use the graph from part (a) to estimate the numbers of hospital employees in 2000,2002 , and 2005 .

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

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt{a-1}, \quad a>1\)

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\ln (x-1)\)

Prescriptions The numbers of prescriptions \(P\) (in millions) filled in the United States from 1998 to 2005 can be modeled by \(P=-11,415+\frac{15,044}{1+e^{-0.2166 t-0.7667}}, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998\. (Source: National Association of Chain Drug Stores) (a) Use a graphing utility to graph \(P\) for the years 1998 to \(2005 .\) (b) Use the graph from part (a) to estimate the numbers of prescriptions filled in 1999,2002 , and 2005 .

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln x-\ln (x+2)=3\)

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

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\ln (x+2)\)

Writing Determine whether \(e=\frac{271,801}{99,990}\). Justify your answer.

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

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \left[\frac{(z-1)^{2}}{z}\right]\)

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\ln x+1\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln x+\ln (x+3)=1\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \left(\frac{x}{\sqrt{x^{2}+1}}\right)\)

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=3 \ln x-1\)