Chapter 5

Solve the equation. $$ \ln x=1-\ln (x+2) $$

Minimum wage In 1971 the minimum wage in the United States was \(\$ 1.60\) per hour. Assuming that the rate of inflation is \(5 \%\) per year, find the equivalent minimum wage in the year 2010 .

Exer. 25-32: Solve the equation without using a calculator. $$ e^{2 x}+2 e^{x}-15=0 $$

Exer. 19-34: Solve the equation. $$ e^{2 \ln x} \equiv 9 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=2-3 x^{2}, x \leq 0 $$

Find an exponential function of the form \(f(x)=b a^{-x}+c\) that has the given horizontal asymptote and \(y\)-intercept and passes through point \(P\). \(y=32 ; \quad y\)-intercept \(212 ; \quad P(2,112)\)

Solve the equation. $$ \ln x=1+\ln (x+1) $$

Land value In 1867 the United States purchased Alaska from Russia for \(\$ 7,200,000\). There is 586,400 square miles of land in Alaska. Assuming that the value of the land increases continuously at \(3 \%\) per year and that land can be purchased at an equivalent price, determine the price of 1 acre in the year 2010 . (One square mile is equivalent to 640 acres.)

Exer. 25-32: Solve the equation without using a calculator. $$ e^{x}+4 e^{-x}=5 $$

Exer. 19-34: Solve the equation. $$ e^{-\ln x}=0.2 $$

Find an exponential function of the form \(f(x)=b a^{-x}+c\) that has the given horizontal asymptote and \(y\)-intercept and passes through point \(P\). \(y=72 ; \quad y\)-intercept \(425 ; \quad P(1,248.5)\)

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=5 x^{2}+2, x \geq 0 $$

Solve the equation. $$ \log _{3}(x-2)=\log _{3} 27-\log _{3}(x-4)-5^{\log _{5} 1} $$

Exer. 33-34: Solve the equation. $$ \log _{3} x-\log _{9}(x+42)=0 $$

Exer. 19-34: Solve the equation. $$ e^{x \ln 3}=27 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=2 x^{3}-5 $$

Elk population One hundred elk, each 1 year old, are introduced into a game preserve. The number \(N(t)\) alive after \(t\) years is predicted to be \(N(t)=100(0.9)^{t}\). Estimate the number alive after (a) 1 year (b) 5 years (c) 10 years

Solve the equation. $$ \log _{2}(x+3)=\log _{2}(x-3)+\log _{3} 9+4^{\log _{4} 3} $$

The effective yield (or effective annual interest rate) for an investment is the simple interest rate that would yield at the end of one year the same amount as is yielded by the compounded rate that is actually applied. Approximate, to the nearest \(0.01 \%\), the effective yield corresponding to an interest rate of \(r \%\) per year compounded (a) quarterly and (b) continuously. $$ r=12 $$

Exer. 33-34: Solve the equation. $$ \log _{4} x+\log _{8} x=1 $$

Exer. 19-34: Solve the equation. $$ e^{x \ln 2}=0.25 $$

Drug dosage A drug is eliminated from the body through urine. Suppose that for an initial dose of 10 milligrams, the amount \(A(t)\) in the body \(t\) hours later is given by \(A(t)=10(0.8)^{t}\) (a) Estimate the amount of the drug in the body 8 hours after the initial dose. (b) What percentage of the drug still in the body is eliminated each hour?

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=-x^{3}+2 $$

Sketch the graph of \(f\). $$ f(x)=\log _{3}(3 x) $$

Probability density function In statistics, the probability density function for the normal distribution is defined by $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-z^{2 / 2}} \quad \text { with } \quad z=\frac{x-\mu}{\sigma} $$ where \(\mu\) and \(\sigma\) are real numbers ( \(\mu\) is the mean and \(\sigma^{2}\) is the variance of the distribution). Sketch the graph of \(f\) for the case \(\sigma=1\) and \(\mu=0\).

Exer. 35-38: Use common logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{10^{x}+10^{-x}}{2} $$

Sketch the graph of \(f\) if \(a=4\) : (a) \(f(x)=\log _{a} x\) (b) \(f(x)=-\log _{a} x\) (c) \(f(x)=2 \log _{a} x\) (d) \(f(x)=\log _{a}(x+2)\) (e) \(f(x)=\left(\log _{a} x\right)+2\) (f) \(f(x)=\log _{a}(x-2)\) (g) \(f(x)=\left(\log _{a} x\right)-2\) (h) \(f(x)=\log _{a}|x|\) (i) \(f(x)=\log _{a}(-x)\) (j) \(f(x)=\log _{a}(3-x)\) (k) \(f(x)=\left|\log _{a} x\right|\) (l) \(f(x)=\log _{1 / a} x\)

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\sqrt{3-x} $$

Drug dosage A drug is eliminated from the body through urine. Suppose that for an initial dose of 10 milligrams, the amount \(A(t)\) in the body \(t\) hours later is given by \(A(t)=10(0.8)^{t}\) (a) Estimate the amount of the drug in the body 8 hours after the initial dose. (b) What percentage of the drug still in the body is eliminated each hour?

Sketch the graph of \(f\). $$ f(x)=\log _{4}(16 x) $$

Exer. 35-38: Use common logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{10^{x}-10^{-x}}{2} $$

Newton's law of cooling According to Newton's law of cooling, the rate at which an object cools is directly proportional to the difference in temperature between the object and the surrounding medium. The face of a household iron cools from \(125^{\circ}\) to \(100^{\circ}\) in 30 minutes in a room that remains at a constant temperature of \(75^{\circ}\). From calculus, the temperature \(f(t)\) of the face after \(t\) hours of cooling is given by \(f(t)=50(2)^{-2 x}+75\). (a) Assuming \(t=0\) corresponds to 1:00 P.M., approximate to the nearest tenth of a degree the temperature of the face at \(2: 00\) P.M., 3:30 P.M., and 4:00 P.M. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 4\).

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\sqrt{4-x^{2}}, 0 \leq x \leq 2 $$

Sketch the graph of \(f\). $$ f(x)=3 \log _{3} x $$

Exer. 35-38: Use common logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}} $$

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\log (x+10) $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\sqrt[3]{x}+1 $$

Sketch the graph of \(f\). $$ f(x)=\frac{1}{3} \log _{3} x $$

Exer. 35-38: Use common logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{10^{x}+10^{-x}}{10^{x}-10^{-x}} $$

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\log (x+100) $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\left(x^{3}+1\right)^{5} $$

Light penetration in an ocean An important problem in oceanography is to determine the amount of light that can penetrate to various ocean depths. The Beer-Lambert law asserts that the exponential function given by \(I(x)=I_{0} c^{x}\) is a model for this phenomenon (see the figure). For a certain location, \(I(x)=10(0.4)^{x}\) is the amount of light (in calories \(\left./ \mathrm{cm}^{2} / \mathrm{sec}\right)\) reaching a depth of \(x\) meters. (a) Find the amount of light at a depth of 2 meters. (b) Sketch the graph of \(I\) for \(0 \leq x \leq 5\).

Sketch the graph of \(f\). $$ f(x)=\log _{3}\left(x^{2}\right) $$

Exer. 39-42: Use natural logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{e^{x}-e^{-x}}{2} $$

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\ln |x| $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=x $$

Sketch the graph of \(f\). $$ f(x)=\log _{2}\left(x^{2}\right) $$

Exer. 39-42: Use natural logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{e^{x}+e^{-x}}{2} $$

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\ln |x-1| $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=-x $$