Chapter 4

Find the exact value of the logarithmic expression without using a calculator.\(\log _{8} \sqrt[4]{8}\)

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln 36.7\)

A grape has a pH of \(3.5\), and baking soda has a pH of \(8.0\). The hydrogen ion concentration of the grape is how many times that of the baking soda?

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\frac{500}{100-e^{x / 2}}=20\)

Find the exact value of the logarithmic expression without using a calculator.\(\ln \frac{1}{\sqrt{e}}\)

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln \sqrt{6}\)

The \(\mathrm{pH}\) of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor?

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\frac{400}{1+e^{-x}}=350\)

Find the exact value of the logarithmic expression without using a calculator.\(\ln \sqrt[4]{e^{3}}\)

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln \sqrt{10}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\frac{3000}{2+e^{2 x}}=2\)

Find the exact value of the logarithmic expression without using a calculator.\(\log _{5} \frac{1}{125}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=3^{x}, g(x)=\log _{3} x\)

Thawing a Package of Steaks You take a three-pound package of steaks out of the freezer at 11 A.M. and place it in the refrigerator. Will the steaks be thawed in time to be grilled at 6 p.m.? Assume that the refrigerator temperature is \(40^{\circ} \mathrm{F}\) and that the freezer temperature is \(0^{\circ} \mathrm{F}\). Use the formula for Newton's Law of Cooling \(t=-5.05 \ln \frac{T-40}{0-40}\) where \(t\) is the time in hours (with \(t=0\) corresponding to 11 A.M.) and \(T\) is the temperature of the package of steaks (in degrees Fahrenheit).

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\frac{119}{e^{6 x}-14}=7\)

Find the exact value of the logarithmic expression without using a calculator.\(\log _{7} \frac{49}{343}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=5^{x}, g(x)=\log _{5} x\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.065}{365}\right)^{365 t}=4\)

Find the exact value of the logarithmic expression without using a calculator.\(\log _{9} \frac{1}{18}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=e^{x}, g(x)=\ln x\)

The demand function for a limited edition comic book is given by \(p=3000\left(1-\frac{5}{5+e^{-0.015 x}}\right)\) (a) Find the price \(p\) for a demand of \(x=75\) units. (b) Find the price \(p\) for a demand of \(x=200\) units. (c) Use a graphing utility to graph the demand function. (d) Use the graph from part (c) to approximate the demand when the price is \(\$ 100\).

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.075}{4}\right)^{4 t}=5\)

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{5} \frac{1}{15}\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=10^{x}, g(x)=\log _{10} x\)

The demand function for a home theater sound system is given by \(p=7500\left(1-\frac{7}{7+e^{-0.003 x}}\right)\) (a) Find the price \(p\) for a demand of \(x=200\) units. (b) Find the price \(p\) for a demand of \(x=900\) units. (c) Use a graphing utility to graph the demand function. (d) Use the graph from part (c) to approximate the demand when the price is \(\$ 400\).

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.10}{12}\right)^{12 t}=2\)

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{7} \sqrt{70}\)

The number of a certain type of bacteria increases according to the model \(P(t)=100 e^{0.01896 t}\) where \(t\) is time (in hours). (a) Find \(P(0)\). (b) Find \(P(5)\). (c) Find \(P(10)\). (d) Find \(P(24)\).

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.0825}{26}\right)^{26 t}=9\)

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{5} \sqrt{75}\)

As a result of a medical treatment, the number of a certain type of bacteria decreases according to the model \(P(t)=100 e^{-0.685 t}\) where \(t\) is time (in hours). (a) Find \(P(0)\). (b) Find \(P(5)\). (c) Find \(P(10)\). (d) Find \(P(24)\).

In Exercises \(61-90\), solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} x=4\)

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{5} \frac{1}{250}\)

Find the present value of amount \(A\) invested at rate \(r\) for \(t\) years, compounded \(n\) times per vear.\(A=\$ 10,000, r=6 \%, t=5\) years, \(n=4\)

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

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{10} \frac{9}{300}\)

Find the present value of amount \(A\) invested at rate \(r\) for \(t\) years, compounded \(n\) times per vear.\(A=\$ 50,000, r=7 \%, t=10\) years, \(n=12\)

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

Use the properties of logarithms to simplify the given logarithmic expression.\(\ln \left(5 e^{6}\right)\)

Find the present value of amount \(A\) invested at rate \(r\) for \(t\) years, compounded \(n\) times per vear.\(A=\$ 20,000, r=8 \%, t=6\) years, \(n=4\)

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

Use the properties of logarithms to simplify the given logarithmic expression.\(\ln \frac{6}{e^{2}}\)

Find the present value of amount \(A\) invested at rate \(r\) for \(t\) years, compounded \(n\) times per vear.\(A=\$ 1,000,000, r=8 \%, t=20\) years, \(n=2\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln 2 x=2.4\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{2}\left(4^{3} \cdot 3^{5}\right)\)

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

Population Growth The population \(P\) of a town increases according to the model \(P(t)=4500 e^{0.0272 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . Use the model to predict the population in each year. (a) 2010 (b) 2012 (c) 2015 (d) 2020

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln 4 x=1\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{3}\left(3^{2} \cdot 4^{2}\right)\)

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