Chapter 3

Sketch the graph of the equation. $$y=\log _{3} x$$

Sketch the graph of the equation. $$y=\log _{1 / 3} x$$

Jane took \(100 \mathrm{mg}\) of a drug in the morning and another \(100 \mathrm{mg}\) of the same drug at the same time the following morning. The amount of the drug in her body \(t\) days after the first dosage was taken is given by $$ A(t)=\left\\{\begin{array}{ll} 100 e^{-1.4 t} & \text { if } 0 \leq t<1 \\ 100\left(1+e^{1.4}\right) e^{-1 . A r} & \text { if } t \geq 1 \end{array}\right. $$ What was the amount of drug in Jane's body immediately after taking the second dose? After 2 days?

Sketch the graph of the equation. $$y=\ln 2 x$$

Sketch the graph of the equation. $$y=\ln \frac{1}{2} x$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$e^{x y}=e^{x} e^{y}$$

Sketch the graphs of the equations on the same coordinate axes. \(y=2^{x}\) and \(y=\log _{2} x\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(x

Use logarithms to solve the equation for \(t\). $$e^{0.4 t}=8$$

Use logarithms to solve the equation for \(t\). $$\frac{1}{3} e^{-3 t}=0.9$$

Use logarithms to solve the equation for \(t\). $$4 e^{t-1}=4$$

Use logarithms to solve the equation for \(t\). $$12-e^{0.4 t}=3$$

Use logarithms to solve the equation for \(t\). $$\frac{50}{1+4 e^{0.2 t}}=20$$

Use logarithms to solve the equation for \(t\). $$\frac{200}{1+3 e^{-0.3 t}}=100$$

Use logarithms to solve the equation for \(t\). $$\frac{A}{1+B e^{t / 2}}=C$$

A function \(f\) has the form \(f(x)=a+b \ln x\). Find \(f\) if it is known that \(f(1)=2\) and \(f(2)=4\).

One reason for the increase in the life span over the years has been the advances in medical technology. The average life span for American women from 1907 through 2007 is given by $$ W(t)=49.9+17.1 \ln t \quad(1 \leq t \leq 6) $$ where \(W(t)\) is measured in years and \(t\) is measured in 20 -yr intervals, with \(t=1\) corresponding to 1907 . a. What was the average life expectancy for women in \(1907 ?\) b. If the trend continues, what will be the average life expectancy for women in \(2027 ?\)

A normal child's systolic blood pressure may be approximated by the function $$ p(x)=m(\ln x)+b $$ where \(p(x)\) is measured in millimeters of mercury, \(x\) is measured in pounds, and \(m\) and \(b\) are constants. Given that \(m=19.4\) and \(b=18\), determine the systolic blood pressure of a child who weighs \(92 \underline{\text { lb. }}\)

On the Richter scale, the magnitude \(R\) of an earthquake is given by the formula $$ R=\log \frac{I}{I_{0}} $$ where \(I\) is the intensity of the earthquake being measured and \(I_{0}\) is the standard reference intensity. a. Express the intensity \(I\) of an earthquake of magnitude \(R=5\) in terms of the standard intensity \(I_{0}\). b. Express the intensity \(I\) of an earthquake of magnitude \(R=8\) in terms of the standard intensity \(I_{0}\). How many times greater is the intensity of an earthquake of magnitude 8 than one of magnitude \(5 ?\) c. In modern times, the greatest loss of life attributable to an earthquake occurred in eastern China in 1976 . Known as the Tangshan earthquake, it registered \(8.2\) on the Richter scale. How does the intensity of this earthquake compare with the intensity of an earthquake of magnitude \(R=5 ?\)

The relative loudness of a sound \(D\) of intensity \(I\) is measured in decibels (db), where $$ D=10 \log \frac{I}{I_{0}} $$ and \(I_{0}\) is the standard threshold of audibility. a. Express the intensity \(I\) of a 30 -db sound (the sound level of normal conversation) in terms of \(I_{0}\). b. Determine how many times greater the intensity of an 80 -db sound (rock music) is than that of a 30 -db sound. c. Prolonged noise above \(150 \mathrm{db}\) causes permanent deafness. How does the intensity of a 150 -db sound compare with the intensity of an 80 -db sound?

Halley's law states that the barometric pressure (in inches of mercury) at an altitude of \(x \mathrm{mi}\) above sea level is approximated by the equation $$ p(x)=29.92 e^{-0.2 x} \quad(x \geq 0) $$ If the barometric pressure as measured by a hot-air balloonist is 20 in. of mercury, what is the balloonist's altitude?

The height (in feet) of a certain kind of tree is approximated by $$ h(t)=\frac{160}{1+240 e^{-0.2 t}} $$ where \(t\) is the age of the tree in years. Estimate the age of an 80 -ft tree.

The temperature of a cup of coffee \(t\) min after it is poured is given by $$ T=70+100 e^{-0.0446 t} $$ where \(T\) is measured in degrees Fahrenheit. a. What was the temperature of the coffee when it was poured? b. When will the coffee be cool enough to drink (say, \(\left.120^{\circ} \mathrm{F}\right) ?\)

The length (in centimeters) of a typical Pacific halibut \(t\) yr old is approximately $$ f(t)=200\left(1-0.956 e^{-0.182}\right) $$ Suppose a Pacific halibut caught by Mike measures \(140 \mathrm{~cm}\). What is its approximate age?

The concentration of a drug in an organ at any time \(t\) (in seconds) is given by $$ x(t)=0.08+0.12 e^{-0.02 t} $$ where \(x(t)\) is measured in grams/cubic centimeter \(\left(\mathrm{g} / \mathrm{cm}^{3}\right)\). a. How long would it take for the concentration of the drug in the organ to reach \(0.18 \mathrm{~g} / \mathrm{cm}^{3}\) ? b. How long would it take for the concentration of the drug in the organ to reach \(0.16 \mathrm{~g} / \mathrm{cm}^{3}\) ?

a. Given that \(2^{x}=e^{k x}\), find \(k\). b. Show that, in general, if \(b\) is a nonnegative real number, then any equation of the form \(y=b^{x}\) may be written in the form \(y=e^{k x}\), for some real number \(k\).

Use the definition of a logarithm to prove a. \(\log _{b} m n=\log _{b} m+\log _{b} n\) b. \(\log _{b} \frac{m}{n}=\log _{b} m-\log _{b} n\)

Use the definition of a logarithm to prove $$ \log _{b} m^{n}=n \log _{b} m $$

Use the definition of a logarithm to prove a. \(\log _{b} 1=0\) b. \(\log _{b} b=1\)