Chapter 4

A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$f(x)=\log (c x)$$

Radioactive Decay A 15-g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t},\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$f(x)=c \log x$$

Sky Diving The velocity of a sky diver \(t\) seconds after jumping is given by \(v(t)=80\left(1-e^{-0.2 t}\right)\). After how many seconds is the velocity 70 ft/s?

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$f(x)=\log _{2}\left(\log _{10} x\right)$$

Fish Population \(A\) small lake is stocked with a certain species of fish. The fish population is modeled by the function $$P=\frac{10}{1+4 e^{-0.8 t}}$$ where \(P\) is the number of fish in thousands and \(t\) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (b) After how many years will the fish population reach 5000 fish?

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$f(x)=\ln (\ln (\ln x))$$

Transparency of a Lake Environmental scientists measure the intensity of light at various depths in a lake to find the "transparency" of the water. Certain levels of transparency are required for the biodiversity of the submerged macrophyte population. In a certain lake the intensity of light at depth \(x\) is given by $$I=10 e^{-0.008 x}$$ where \(I\) is measured in lumens and \(x\) in feet. (a) Find the intensity \(I\) at a depth of \(30 \mathrm{ft}\). (b) At what depth has the light intensity dropped to \(I=5 ?\)

(a) Find the inverse of the function \(f(x)=\frac{2^{x}}{1+2^{x}}\). (b) What is the domain of the inverse function?

Atmospheric Pressure Atmospheric pressure \(P\) (in kilopascals, kPa) at altitude \(h\) (in kilometers, \(\mathrm{km}\) ) is governed by the formula $$\ln \left(\frac{P}{P_{0}}\right)=-\frac{h}{k}$$ where \(k=7\) and \(P_{0}=100 \mathrm{kPa}\) are constants. (a) Solve the equation for \(P\). (b) Use part (a) to find the pressure \(P\) at an altitude of \(4 \mathrm{km}\)

A spectrophotometer measures the concentration of a sample dissolved in water by shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter) is found by using the formula $$C=-2500 \ln \left(\frac{I}{I_{0}}\right)$$ where \(I_{0}\) is the intensity of the incident light and \(I\) is the intensity of light that emerges. Find the concentration of the substance if the intensity \(I\) is \(70 \%\) of \(I_{0}\).

Cooling an Engine Suppose you're driving your car on a cold winter day \(\left(20^{\circ} \mathrm{F}\) outside ) and the engine overheats (at \right. about \(220^{\circ} \mathrm{F}\) ). When you park, the engine begins to cool down. The temperature \(T\) of the engine \(t\) minutes after you park satisfies the equation $$\ln \left(\frac{T-20}{200}\right)=-0.11 t$$ (a) Solve the equation for \(T\). (b) Use part (a) to find the temperature of the engine after \(20 \min (t=20)\)

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon-14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$A=-8267 \ln \left(\frac{D}{D_{0}}\right)$$ Find the age of an object if the amount \(D\) of carbon- 14 that remains in the object is \(73 \%\) of the original amount \(D_{0}\).

Electric Circuits An electric circuit contains a battery that produces a voltage of 60 volts \((\mathrm{V}),\) a resistor with a resistance of 13 ohms \((\Omega),\) and an inductor with an inductance of 5 henrys \((\mathrm{H}),\) as shown in the figure. Using calculus, it can be shown that the current \(I=I(t)\) (in amperes, A) \(t\) seconds after the switch is closed is \(I=\frac{60}{13}\left(1-e^{-13 / 5}\right)\) (a) Use this equation to express the time \(t\) as a function of the current \(I\) (b) After how many seconds is the current \(2 \mathrm{A} ?\)

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time \(t\) (in hours) required for the colony to grow to \(N\) bacteria is given by $$t=3 \frac{\log (N / 50)}{\log 2}$$ Find the time required for the colony to grow to a million bacteria.

Learning Curve A learning curve is a graph of a function \(P(t)\) that measures the performance of someone learning a skill as a function of the training time \(t\). At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value \(M\), the rate of learning decreases. It has been found that the function $$P(t)=M-C e^{-k t}$$ where \(k\) and \(C\) are positive constants and \(C

The time required to double the amount of an investment at an interest rate \(r\) compounded continuously is given by $$t=\frac{\ln 2}{r}$$ Find the time required to double an investment at \(6 \%, 7 \%\) and \(8 \%\).

Estimating a Solution Without actually solving the equation, find two whole numbers between which the solution of \(9^{x}=20\) must lie. Do the same for \(9^{x}=100 .\) Explain how you reached your conclusions.

Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge \(C_{0}\). The time (in hours) required to charge a fully discharged battery to a charge \(C\) is given by $$t=-k \ln \left(1-\frac{C}{C_{0}}\right)$$ where \(k\) is a positive constant that depends on the battery. For a certain battery, \(k=0.25 .\) If this battery is fully discharged, how long will it take to charge to \(90 \%\) of its maximum charge \(C_{0} ?\)

A Surprising Equation Take logarithms to show that the equation $$x^{1 / \log x}=5$$ has no solution. For what values of \(k\) does the equation $$x^{1 / \log x}=k$$ have a solution? What does this tell us about the graph of the function \(f(x)=x^{1 / \log x ?}\) Confirm your answer using a graphing device.

The difficulty in "acquiring a target" (such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fitts's Law, the index of difficulty (ID) is given by $$\mathrm{ID}=\frac{\log (2 A / W)}{\log 2}$$ where \(W\) is the width of the target and \(A\) is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 mm wide to clicking on one that is 100 mm wide. In each case, assume that the mouse is 100 mm from the icon. (Figure cant copy)

Disguised Equations Each of these equations can be transformed into an equation of linear or quadratic type by applying the hint. Solve each equation. (a) \((x-1)^{\log (x-1)}=100(x-1) \quad\) [Take log of each side.] (b) \(\left.\log _{2} x+\log _{4} x+\log _{8} x=11 \quad \text { [Change all logs to base } 2\right]\) \(\begin{array}{lll}\text { (c) } 4^{x}-2^{x+1}=3 & \text { [Write as a quadratic in } 2^{x} \text { .] }\end{array}\)

Suppose that the graph of \(y=2^{x}\) is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 mi. (b) If the graph of \(y=\log _{2} x\) is drawn on the same set of axes, how far to the right of the origin do we have to go before the height of the curve reaches 2 ft?

Which is larger, \(\log _{4} 17\) or \(\log _{5} 24 ?\) Explain your reasoning.

Compare log 1000 to the number of digits in 1000 . Do the same for 10,000 . How many digits does any number between 1000 and 10,000 have? Between what two values must the common logarithm of such a number lie? Use your observations to explain why the number of digits in any positive integer \(x\) is \(\|\log x\|+1 .\) (The symbol \(\|n\|\) is the greatest integer function defined in Section 2.2 . ) How many digits does the number \(2^{100}\) have?