Chapter 5

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log _{3} 5 x=10$$

A student insists that \(\log \left(\frac{x}{y}\right)\) and \(\frac{\log x}{\log y}\) are equal. How could you convince the student otherwise?

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ g^{-1}(0) $$

Thickness of Runways Heavier aircraft require runways with thicker pavement for landings and takeoffs. A pavement 6 inches thick can accommodate an aircraft weighing \(80,000\) pounds, whereas a 12 -inch-thick pavement is necessary for a \(350,000\) -pound plane. The relation between pavement thickness \(t\) in inches and gross weight \(W\) in thousands of pounds can be modeled by \(W=C a^{t} .\) (a) Find values for \(C\) and \(a\) (b) How heavy an airplane can a 9 -inch-thick runway accommodate? (c) What is the minimum thickness for a \(242,000\) -pound plane?

Solve each equation. Approximate answers to four decimal places when appropriate. $$5 \ln (2 x)+6=12$$

Filters Impurities in water are frequently removed using filters. Suppose that a 1 -inch filter allows \(10 \%\) of the impurities to pass through it. The other \(90 \%\) is trapped in the filter. (a) Find a formula in the form \(f(x)=100 a^{x}\) that calculates the percentage of impurities passing through \(x\) inches of this type of filter. (b) Use \(f(x)\) to estimate the percentage of impurities passing through 2.3 inches of the filter.

Solve each equation. Approximate answers to four decimal places when appropriate. $$16-4 \ln 3 x=2$$

Revenue, Cost, and Profit Suppose that it costs \(\$ 150,000\) to produce a master disc for a music video and \(\$ 1.50\) to produce each copy. (a) Write a cost function \(C\) that outputs the cost of producing the master disc and x copies. (b) If the music videos are sold for \(\$ 6.50\) each, find a function \(R\) that outputs the revenue received from selling x music videos. What is the revenue from selling 8000 videos? (c) Assuming that the master disc is not sold, find a function \(\mathbf{P}\) that outputs the profit from selling \(\mathbf{x}\) music videos. What is the profit from selling \(40,000\) videos? (d) How many videos must be sold to break even? That is, how many videos must be sold for the revenue to equal the cost?

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ \left(g^{-1} \circ g^{-1}\right)(2) $$

Trains The faster a locomotive travels, the more horsepower is needed. The formula \(H(x)=0.157(1.033)^{x}\) calculates this horsepower for a level track. The input \(x\) is in miles per hour and the output \(H(x)\) is the horsepower required per ton of cargo. (a) Evaluate \(H(30)\) and interpret the result. (b) Determine the horsepower needed to move a 5000 ton train 30 miles per hour. (c) Some types of locomotives are rated for 1350 horsepower. How many locomotives of this type would be needed in part (b)?

Solve each equation. Approximate answers to four decimal places when appropriate. $$9-3 \log _{4} 2 x=3$$

Converting Units There are 36 inches in a yard and 2.54 centimeters in an inch. (a) Write a function I that converts \(x\) yards to inches. (b) Write a function \(C\) that converts \(x\) inches to centimeters. (c) Express a function \(F\) that converts \(x\) yards to centimeters as a composition of two functions. (d) Write a formula for \(\mathbf{F}\).

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ \left(g \circ f^{-1}\right)(5) $$

Survival of Reindeer For all types of animals, the percentage that survive into the next year decreases. In one study, the survival rate of a sample of reindeer was modeled by \(S(t)=100(0.999993)^{t^{3}} .\) The function \(S\) outputs the percentage of reindeer that survive \(t\) years. (a) Bvaluate \(S(4)\) and \(S(15)\). Interpret the results. (b) Graph \(S\) in \([0,15,5]\) by \([0,110,10]\). Interpret the graph. Does the graph have a horizontal asymptote?

Solve each equation. Approximate answers to four decimal places when appropriate. $$7 \log _{6}(4 x)+5=-2$$

Converting Units There are 4 quarts in 1 gallon, 4 cups in 1 quart, and 16 tablespoons in 1 cup. (a) Write a function \(Q\) that converts \(x\) gallons to quarts. (b) Write a function \(C\) that converts \(x\) quarts to cups. (c) Write a function \(\mathbf{T}\) that converts \(\mathbf{x}\) cups to tablespoons. (d) Express a function \(F\) that converts \(x\) gallons to tablespoons as a composition of three functions. (e) Write a formula for \(\mathbf{F}\).

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ \left(f^{-1} \circ g\right)(4) $$

Explain how linear and exponential functions differ. Give examples.

Graph \(f\) and state its domain. $$f(x)=\log (x+1)$$

Graph \(f\) and state its domain. $$f(x)=\log (x-3)$$

Graph \(f\) and state its domain. $$f(x)=\ln (-x)$$

Graph \(f\) and state its domain. $$f(x)=\ln \left(x^{2}+1\right)$$

Circular Wave A marble is dropped into a lake, resulting in a circular wave whose radius increases at a rate of 6 inches per second. Write a formula for \(C\) that gives the circumference of the circular wave in inches after \(t\) seconds.

Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 2} x$$

Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 3} x$$

Geometry The surface area of a cone (excluding the bottom) is given by \(S=\pi r \sqrt{r^{2}+h^{2}},\) where \(r\) is its radius and \(h\) is its height, as shown in the figure. If the height is twice the radius, write a formula for \(S\) in terms of \(r\) (GRAPH CANT COPY)

Complete the following. (a) Graph \(y=f(x), y=f^{-1}(x),\) and \(y=x\) (b) Determine the intervals where \(f\) and \(f^{-1}\) are increasing or decreasing. $$f(x)=\log _{3} x$$

Equilateral Triangle The area of an equilateral triangle with sides of length \(s\) is given by $$ A(s)=\frac{\sqrt{3}}{4} s^{2} $$ (a) Find \(A(4 s)\) and interpret the result. (b) Find \(A(s+2)\) and interpret the result.

Complete the following. (a) Graph \(y=f(x), y=f^{-1}(x),\) and \(y=x\) (b) Determine the intervals where \(f\) and \(f^{-1}\) are increasing or decreasing. $$f(x)=\log _{1 / 2} x$$

Surface Area of a Balloon The surface area \(A\) of a balloon with radius \(r\) is given by \(A(r)=4 \pi r^{2} .\) Suppose that the radius of the balloon increases from \(r\) to \(r+h\) where \(h\) is a small positive number. (a) Find \(A(r+h)-A(r) .\) Interpret your answer. (b) Evaluate your expression in part (a) when \(r=3\) and \(h=0.1 .\) Then evaluate it for \(r=6\) and \(h=0.1\) (c) If the radius of the balloon increases by \(0.1,\) does the surface area always increase by a fixed amount or does the amount depend on the value of \(r ?\)

Change each equation to its equivalent logarithmic form. (a) \(7^{4 x}=4\) (b) \(e^{x}=7\) (c) \(c^{x}=b\)

Change each equation to its equivalent logarithmic form. (a) \(5^{2 x}=9\) (b) \(b^{x}=a\) (c) \(d^{2 x}=b\)

Change each equation to its equivalent exponential form. (a) \(\log _{8} x=3\) (b) \(\log _{9}(2+x)=5\) (c) \(\log _{k} b=c\)

Decibels (Refer to Example 2.) Use the formula \(D(x)=10 \log \left(10^{16} x\right)\) to determine the decibels when the intensity of a sound is \(x=10^{-1 / 2}\) watt per square centimeter.

Energy of a Falling Object \(\mathbf{A}\) ball with mass \(m\) is dropped from an initial height of \(h_{0}\) and lands with a final velocity of \(v_{f}\). The kinetic energy of the ball is \(K(v)=\frac{1}{2} m v^{2},\) where \(v\) is its velocity, and the potential energy of the ball is \(P(h)=m g h,\) where \(h\) is its height and \(g\) is a constant. (a) Show that \(P\left(h_{0}\right)=K\left(v_{f}\right) .\) (Hint: \(v_{f}=\sqrt{2 g h_{0}}\) ) (b) Interpret your result from part (a).

Sphere The volume \(V\) of a sphere with radius \(r\) is given by \(V=\frac{4}{3} \pi r^{3},\) and the surface area \(S\) is given by \(S=4 \pi r^{2} .\) Show that \(V=\frac{4}{3} \pi\left(\frac{s}{4 \pi}\right)^{3 / 2}\)

Applying Concepts Show that the sum of two linear functions is a linear function.

Show that if \(f\) and \(g\) are odd functions, then the composition \(g \circ f\) is also an odd function.

Runvay Length There is a relation between an airplane's weight \(x\) and the runway length \(L\) required for takeoff. For some airplanes the minimum runway length \(L\) in thousands of feet may be modeled by \(L(x)=3 \log x,\) where \(x\) is measured in thousands of pounds. (Sourcet. I. Haefner, Introduction to Thangortation Systems.) (a) Graph \(L\) for \(0

Let \(f(x)=k\) and \(g(x)=a x+b,\) where \(k, a,\) and \(b\) are constants. (a) Find \((f \circ g)(x) .\) What type of function is \(f \circ g ?\) (b) Find \((g \circ f)(x) .\) What type of function is \(g \circ f ?\)

Height and Weight The formula \(W=\frac{25}{7} h-\frac{800}{7}\) approximates the recommended minimum weight for a person \(h\) inches tall, where \(62 \leq h \leq 76.\) (a) What is the recommended minimum weight for someone 70 inches tall? (b) Does \(W\) represent a one-to-one function? (c) Find a formula for the inverse. (d) Evaluate the inverse for 150 pounds and interpret the result. (e) What does the inverse compute?

Show that if \(f(x)=a x+b\) and \(g(x)=c x+d,\) then \((g \circ f)(x)\) also represents a linear function. Find the slope of the graph of \((g \circ f)(x)\)

Planetary Orbils \(\quad\) The formula \(T(x)=x^{3 / 2}\) calculates the time in years that it takes a planet to orbit the sun if the planet is \(x\) times farther from the sun than Earth is. (a) Find the inverse of \(T\). (b) What does the inverse of \(T\) calculate?

Acid Rain Air pollutants frequently cause acid rain. A measure of the acidity is \(\mathrm{pH}\), which ranges between 1 and \(14 .\) Pure water is neutral and has a \(\mathrm{pH}\) of \(7 .\) Acidic solutions have a \(\mathrm{pH}\) less than 7 , whereas alkaline solutions have a pH greater than \(7 .\) A pH value can be computed by \(\mathrm{pH}=-\log x,\) where \(x\) represents the hydrogen ion concentration in moles per liter. In rural areas of Europe, rainwater typically has \(x=10^{-4.7}\) (a) Find its \(\mathrm{pH}\). (b) Seawater has a \(\mathrm{pH}\) of \(8.2 .\) How many times greater is the hydrogen ion concentration in rainwater from rural Europe than in seawater?

Writing about Mathematics Describe differences between \((f g)(x)\) and \((f \circ g)(x)\) Give examples.

Converting Units \(\quad\) The tables represent a function \(F\) that converts yards to feet and a function \(Y\) that converts miles to yards. Evaluate each expression and interpret the results. $$ \begin{array}{rlllll} x \text { (yd) } & 1760 & 3520 & 5280 & 7040 & 8800 \\ F(x) \text { (ft) } & 5280 & 10,560 & 15,840 & 21,120 & 26,400 \end{array} $$ $$ \begin{array}{cccccc} x(m i) & 1 & 2 & 3 & 4 & 5 \\ Y(x)(y d) & 1760 & 3520 & 5280 & 7040 & 8800 \end{array} $$ (a) \((F \circ Y)(2)\) (b) \(F^{-1}(26,400)\) (c) \(\left(Y^{-1} \cdot F^{-1}\right)(21,120)\)

Describe differences between \((f \circ g)(x)\) and \((g \circ f)(x)\) Give examples.

Converting Units \(\quad\) The tables represent a function \(C\) that converts tablespoons to cups and a function \(Q\) that converts cups to quarts. Evaluate each expression and interpret the results. $$ \begin{array}{ccccc} x(\text { tbsp }) & 32 & 64 & 96 & 128 \\ C(x)(c) & 2 & 4 & 6 & 8 \end{array} $$ $$ \begin{array}{ccccc} x(c) & 2 & 4 & 6 & 8 \\ Q(x)(q t) & 0.5 & 1 & 1.5 & 2 \end{array} $$ (a) \((Q \circ C)(96)\) (b) \(Q^{-1}(2)\) (c) \(\left(C^{-1} \cdot Q^{-1}\right)(1.5)\)

Hurricanes Hurricanes are some of the largest storms on earth. They are very low pressure areas with diameters of over 500 miles. The barometric air pressure in inches of mercury at a distance of \(x\) miles from the eye of a severe hurricane is modeled by the formula \(f(x)=0.48 \ln (x+1)+27\) (a) Evaluate \(f(0)\) and \(f(100)\). Interpret the results. (b) Graph \(f\) in \([0,250,50]\) by \([25,30,1] .\) Describe how air pressure changes as one moves away from the eye of the hurricane. (c) At what distance from the eye of the hurricane is the air pressure 28 inches of mercury?

Air Pollution Tiny particles suspended in the air are necessary for clouds to form. Experts believe that air pollutants may cause an increase in cloud cover. From 1930 to 1980 the percentage of cloud cover over the world's oceans was monitored. The formula \(f(x)=0.06(x-1930)+62.5\) for \(1930 \leq x \leq 1980\) approximates this percentage. (a) Evaluate the expressions \(f(1930)\) and \(f(1980)\) How did the amount of cloud cover over the oceans change during this 50 -year period? (b) What does \(f^{-1}(x)\) compute? (c) Use part (a) to evaluate \(f^{-1}(62.5)\) and \(f^{-1}(65.5)\) (d) Find \(f^{-1}(x)\)