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Review Exercises

Concepts and Vocabulary

In Exercises 1–10, state whether the given statement is true or false.

The function $f (x) = a^{x}$ $f (x) = a^{x}$ is exponential if $a > 0 .$ $a > 0 .$
The graph of $f (x) = 4^{x}$ $f (x) = 4^{x}$ approaches the x-axis as $x \to - \infty .$ $x \to - \infty .$
The domain of $f (x) = \log (2 - x)$ $f (x) = \log (2 - x)$ is $(- \infty, 2] .$ $(- \infty, 2] .$
The equation $u = 10^{v}$ $u = 10^{v}$ means that $\log u = v .$ $\log u = v .$
The inverse of $f (x) = \ln x$ $f (x) = \ln x$ is $g (x) = e^{x} .$ $g (x) = e^{x} .$
The graph of $y = a^{x} (a > 0, a \neq 1)$ $y = a^{x} (a > 0, a \neq 1)$ always contains the points (0, 1) and (1, a).
$\ln (M + N) = \ln M + \ln N$ $\ln (M + N) = \ln M + \ln N$
$\log \sqrt{300} = 1 + \frac{1}{2} \log 3$ $\log \sqrt{300} = 1 + \frac{1}{2} \log 3$
The functions $f (x) = 2^{- x}$ $f (x) = 2^{- x}$ and $g (x) = {(\frac{1}{2})}^{x}$ $g (x) = {(\frac{1}{2})}^{x}$ have the same graph.
$\ln u = \frac{\log u}{\log e}$ $\ln u = \frac{\log u}{\log e}$

Building Skills

In Exercises 11–18, match the function with its graph in (a)–(h).

1. $f_{1} (x) = \log_{2} x$ $f_{1} (x) = \log_{2} x$
2. $f_{2} (x) = 2^{x}$ $f_{2} (x) = 2^{x}$
3. $f_{3} (x) = \log_{2} (3 - x)$ $f_{3} (x) = \log_{2} (3 - x)$
4. $f_{4} (x) = \log_{1 / 2} (x - 1)$ $f_{4} (x) = \log_{1 / 2} (x - 1)$
5. $f_{5} (x) = - \log_{2} x$ $f_{5} (x) = - \log_{2} x$
6. $f_{6} (x) = {(\frac{1}{2})}^{x}$ $f_{6} (x) = {(\frac{1}{2})}^{x}$
7. $f_{7} (x) = 3 - 2^{- x}$ $f_{7} (x) = 3 - 2^{- x}$
8. $f_{8} (x) = \frac{6}{1 + 2 e^{- x}}$ $f_{8} (x) = \frac{6}{1 + 2 e^{- x}}$

In Exercises 19–30, graph each function using transformations on an appropriate graph. Determine the domain, range, and asymptotes (if any).

$f (x) = 2^{- x}$ $f (x) = 2^{- x}$
$g (x) = 2^{- 0.5 x}$ $g (x) = 2^{- 0.5 x}$
$h (x) = 3 + 2^{- x}$ $h (x) = 3 + 2^{- x}$
$f (x) = 5^{| x |}$ $f (x) = 5^{| x |}$
$g (x) = 5^{- | x |}$ $g (x) = 5^{- | x |}$
$h (x) = e^{- x + 1}$ $h (x) = e^{- x + 1}$
$f (x) = \ln (- x)$ $f (x) = \ln (- x)$
$g (x) = 2 \ln | x |$ $g (x) = 2 \ln | x |$
$h (x) = 2 \ln (x - 1)$ $h (x) = 2 \ln (x - 1)$
$f (x) = 2 - {(\frac{1}{2})}^{x}$ $f (x) = 2 - {(\frac{1}{2})}^{x}$
$g (x) = 2 - \ln (- x)$ $g (x) = 2 - \ln (- x)$
$h (x) = 3 + 2 \ln (5 + x)$ $h (x) = 3 + 2 \ln (5 + x)$

In Exercises 31–34, sketch the graph of the given function using the following two steps:

Find the intercepts.
Find the end behavior of f.

$f (x) = 3 - 2 e^{- x}$ $f (x) = 3 - 2 e^{- x}$
$f (x) = \frac{5}{2 + 3 e^{- x}}$ $f (x) = \frac{5}{2 + 3 e^{- x}}$
$f (x) = e^{- x^{2}}$ $f (x) = e^{- x^{2}}$
$f (x) = 3 - \frac{6}{1 + 2 e^{- x}}$ $f (x) = 3 - \frac{6}{1 + 2 e^{- x}}$

In Exercises 35–38, find a and k and then evaluate the function.

Let $f (x) = a (2^{k x})$ $f (x) = a (2^{k x})$ with $f (0) = 10$ $f (0) = 10$ and $f (3) = 640 .$ $f (3) = 640 .$ Find f(2).
Let $f (x) = 50 - a (5^{k x})$ $f (x) = 50 - a (5^{k x})$ with $f (0) = 10$ $f (0) = 10$ and $f (2) = 0 .$ $f (2) = 0 .$ Find f(1).
Let $f (x) = \frac{4}{1 + a e^{- k x}}$ $f (x) = \frac{4}{1 + a e^{- k x}}$ with $f (0) = 1$ $f (0) = 1$ and $f (3) = \frac{1}{2} .$ $f (3) = \frac{1}{2} .$ Find f(4).
Let $f (x) = 6 - \frac{3}{1 + a e^{- k x}}$ $f (x) = 6 - \frac{3}{1 + a e^{- k x}}$ with $f (0) = 5$ $f (0) = 5$ and $f (4) = 4 .$ $f (4) = 4 .$ Find f(10).

In Exercises 39 and 40, find an exponential function of the form $f (x) = c a^{x}$ $f (x) = c a^{x}$ with the given graph.

In Exercises 41 and 42, find a logarithmic function of the form $y = \log_{a} (x - c)$ $y = \log_{a} (x - c)$ with the given graph.

Start with the graph of $y = 2^{x} .$ $y = 2^{x} .$ Find an equation of the graph that results when you follow the sequence:
1. shift the graph right 1 unit, shift up 3 units, and reflect about the x-axis.
2. shift the graph right 1 unit, reflect about the x-axis, and shift up 3 units.
Start with the graph of $y = \ln x .$ $y = \ln x .$ Find an equation of the graph that results when you follow the sequence:
1. shift the graph left 1 unit, stretch horizontally by a factor of 2, and compress vertically by a factor of 3.
2. compress horizontally to $\frac{1}{3},$ $\frac{1}{3},$ shift left by 1 unit, and reflect about the y-axis.

In Exercises 45–48, write each logarithm in expanded form.

$\ln (x y^{2} z^{3})$ $\ln (x y^{2} z^{3})$
$\log (x^{3} \sqrt{y - 1})$ $\log (x^{3} \sqrt{y - 1})$
$\ln [\frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 3)}^{2}}]$ $\ln [\frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 3)}^{2}}]$
$\ln \sqrt{\frac{x^{3} + 5}{x^{3} - 7}}$ $\ln \sqrt{\frac{x^{3} + 5}{x^{3} - 7}}$

In Exercises 49–54, write y as a function of x.

$\ln y = \ln x + \ln 3$ $\ln y = \ln x + \ln 3$
$\ln y = \ln (C) + k x;$ $\ln y = \ln (C) + k x;$ C and k are constants.
$\ln y = \ln (x - 3) - \ln (y + 2)$ $\ln y = \ln (x - 3) - \ln (y + 2)$
$\ln y = \ln x - \ln (x^{2} y) - 2 \ln y$ $\ln y = \ln x - \ln (x^{2} y) - 2 \ln y$
$\ln y = \frac{1}{2} \ln (x - 1) + \frac{1}{2} \ln (x + 1) - \ln (x^{2} + 1)$ $\ln y = \frac{1}{2} \ln (x - 1) + \frac{1}{2} \ln (x + 1) - \ln (x^{2} + 1)$
$\ln (y - 1) = \frac{1}{x} + \ln y$ $\ln (y - 1) = \frac{1}{x} + \ln y$

In Exercises 55–78, solve each equation.

$3^{x} = 81$ $3^{x} = 81$
$5^{x - 1} = 625$ $5^{x - 1} = 625$
$2^{x^{2} + 2 x} = 16$ $2^{x^{2} + 2 x} = 16$
$3^{x^{2} - 6 x + 8} = 1$ $3^{x^{2} - 6 x + 8} = 1$
$3^{x} = 23$ $3^{x} = 23$
$2^{x - 1} = 5.2$ $2^{x - 1} = 5.2$
$273^{x} = 19$ $273^{x} = 19$
$27 = 9^{x} \cdot 3^{x^{2}}$ $27 = 9^{x} \cdot 3^{x^{2}}$
$3^{2 x} = 7^{x}$ $3^{2 x} = 7^{x}$
$2^{x + 4} = 3^{x + 1}$ $2^{x + 4} = 3^{x + 1}$
${(1.7)}^{3 x} = 3^{2 x - 1}$ ${(1.7)}^{3 x} = 3^{2 x - 1}$
$3 (2^{x + 5}) = 5 (7^{2 x - 3})$ $3 (2^{x + 5}) = 5 (7^{2 x - 3})$
$\log_{3} (x + 2) - \log_{3} (x - 1) = 1$ $\log_{3} (x + 2) - \log_{3} (x - 1) = 1$
$\log_{3} (x + 12) - \log_{3} (x + 4) = 2$ $\log_{3} (x + 12) - \log_{3} (x + 4) = 2$
$\log_{2} (x + 2) + \log_{2} (x + 4) = 3$ $\log_{2} (x + 2) + \log_{2} (x + 4) = 3$
$\log_{5} (3 x + 7) + \log_{5} (x - 5) = 2$ $\log_{5} (3 x + 7) + \log_{5} (x - 5) = 2$
$\log_{5} (x^{2} - 5 x + 6) - \log_{5} (x - 2) = 1$ $\log_{5} (x^{2} - 5 x + 6) - \log_{5} (x - 2) = 1$
$\log_{3} (x^{2} - x - 6) - \log_{3} (x - 3) = 1$ $\log_{3} (x^{2} - x - 6) - \log_{3} (x - 3) = 1$
$\log_{6} (x - 2) + \log_{6} (x + 1) = \log_{6} (x + 4) + \log_{6} (x - 3)$ $\log_{6} (x - 2) + \log_{6} (x + 1) = \log_{6} (x + 4) + \log_{6} (x - 3)$
$\ln (x - 2) - \ln (x + 2) = \ln (x - 1) - \ln (2 x + 1)$ $\ln (x - 2) - \ln (x + 2) = \ln (x - 1) - \ln (2 x + 1)$
$2 \ln 3 x = 3 \ln x$ $2 \ln 3 x = 3 \ln x$
$2 \log x = \ln e$ $2 \log x = \ln e$
$2^{x} - 8 \cdot 2^{- x} - 7 = 0$ $2^{x} - 8 \cdot 2^{- x} - 7 = 0$
$3^{x} - 24 \cdot 3^{- x} = 10$ $3^{x} - 24 \cdot 3^{- x} = 10$

In Exercises 79–82, solve each inequality.

$3 {(0.2)}^{x} + 5 \leq 20$ $3 {(0.2)}^{x} + 5 \leq 20$
$- 4 {(0.2)}^{x} + 15 < 6$ $- 4 {(0.2)}^{x} + 15 < 6$
$\log (2 x + 7) < 2$ $\log (2 x + 7) < 2$
$\ln (3 x + 5) \leq 1$ $\ln (3 x + 5) \leq 1$

Applying the Concepts

Doubling your money. How much time is required for a $1000 investment to double in value if interest is earned at the rate of 6.25% compounded annually?
Tripling your money. How much time (to the nearest month) is required to triple your money if the interest rate is 100% compounded continuously?
Half-life. The half-life of a certain radioactive substance is 20 hours. How long does it take for this substance to fall to 25% of its original value?
Plutonium-210. Find the half-life of plutonium-210 assuming that its decay equation is $Q = Q_{0} e^{- 5 \cdot 10^{- 3} t},$ $Q = Q_{0} e^{- 5 \cdot 10^{- 3} t},$ where t is in days.
Comparing rates. You have $7000 to invest for seven years. Which investment will provide the greater return, 5% compounded yearly or 4.75% compounded monthly?
Investment growth. How long will it take $8000 to grow to $20,000 if the rate of interest is 7% compounded continuously?
Population. The formula $P (t) = 33 e^{0.003 t}$ $P (t) = 33 e^{0.003 t}$ models the population of Canada, in millions, t years after 2007.
1. Estimate the population of Canada in 2017.
2. According to this model, when will the population of Canada be 60 million?
House appreciation. The formula $C (t) = 100 + 25 e^{0.03 t}$ $C (t) = 100 + 25 e^{0.03 t}$ models the average cost of a house in Sometown, USA, t years after 2000. The cost is expressed in thousands of dollars.
1. Sketch the graph of $y = C (t) .$ $y = C (t) .$
2. Estimate the average cost in 2010.
3. According to this model, when will the average cost of a house in Sometown be $250,000?
Drug concentration. An experimental drug was injected into the bloodstream of a rat. The concentration C(t) of the drug (in micrograms per milliliter of blood) after t hours was modeled by the function

$C (t) = 0.3 e^{- 0.47 t}, 0.5 \leq t \leq 10 .$ $C (t) = 0.3 e^{- 0.47 t}, 0.5 \leq t \leq 10 .$
1. Graph the function $y = C (t)$ $y = C (t)$ for $0.5 \leq t \leq 10 .$ $0.5 \leq t \leq 10 .$
2. When will the concentration of the drug be 0.029 microgram per milliliter? $(1 microgram = 10^{- 6} gram)$ $(1 microgram = 10^{- 6} gram)$
X-ray intensity. X-ray technicians are shielded by a wall insulated with lead. The equation $x = \frac{1}{152} \log (\frac{I_{0}}{I})$ $x = \frac{1}{152} \log (\frac{I_{0}}{I})$ measures the thickness x (in centimeters) of the lead insulation required to reduce the initial intensity $I_{0}$ $I_{0}$ of X-rays to the desired intensity I.
1. What thickness of lead is required to reduce the intensity of X-rays to one-tenth their initial intensity?
2. What thickness of lead is required to reduce the intensity of X-rays to $\frac{1}{40}$ $\frac{1}{40}$ their initial intensity?
3. How much is the intensity reduced if the lead insulation is 0.2 millimeter thick?
Cooling tea. Chai (tea) is made by adding boiling water $(212 ° F)$ $(212 ° F)$ to the chai mix. Suppose you make chai in a room with the air temperature at $75 ° F .$ $75 ° F .$ According to Newton’s Law of Cooling, the temperature of the chai t minutes after it is boiled is given by a function of the form $f (t) = 75 + a e^{- k t} .$ $f (t) = 75 + a e^{- k t} .$ After one minute, the temperature of the chai falls to $200 ° F .$ $200 ° F .$ How long will it take for the chai to be drinkable at $150 ° F ?$ $150 ° F ?$
Spread of influenza. Approximately $P (t) = \frac{6}{1 + 5 e^{- 0.7 t}}$ $P (t) = \frac{6}{1 + 5 e^{- 0.7 t}}$ thousand people caught a new form of influenza within t weeks of its outbreak.
1. Sketch the graph of $y = P (t) .$ $y = P (t) .$
2. How many people had the disease initially?
3. How many people contracted the disease within four weeks?
4. If the trend continues, how many people in all will contract the disease?
Population density. The population density x miles from the center of a town called Greenville is approximated by the equation $D (x) = 5 e^{0.08 x},$ $D (x) = 5 e^{0.08 x},$ in thousands of people per square mile.
1. What is the population density at the center of Greenville?
2. What is the population density 5 miles from the center of Greenville?
3. Approximately how far from the center of Greenville would the density be 15,000 people per square mile?
Population. In 2000, the population of the United States was 280 million and the number of vehicles was 200 million. If the population of the United States is growing at the rate of 1% per year while the number of vehicles is growing at the rate of 3%, in what year will there be an average of one vehicle per person?
Light intensity. The Bouguer–Lambert Law states that the intensity I of sunlight filtering down through water at a depth x (in meters) decreases according to the exponential decay function $I = I_{0} e^{- k x},$ $I = I_{0} e^{- k x},$ where $I_{0}$ $I_{0}$ is the intensity at the surface and $k > 0$ $k > 0$ is an absorption constant that depends on the murkiness of the water. Suppose the absorption constant of Carrollwood Lake was experimentally determined to be $k = 0.73 .$ $k = 0.73 .$ How much light has been absorbed by the water at a depth of 2 meters?
Signal strength. The strength of a TV signal usually fades due to a damping effect of cable lines. If $I_{0}$ $I_{0}$ is the initial strength of the signal, then its strength I at a distance x miles is measured by the formula $I = I_{0} e^{- k x},$ $I = I_{0} e^{- k x},$ where $k > 0$ $k > 0$ is a damping constant that depends on the type of wire used. Suppose the damping constant has been measured experimentally to be $k = 0.003 .$ $k = 0.003 .$ What percent of the signal is lost at a distance of 10 miles? 20 miles?
Walking speed in a city. In 1976, Marc and Helen Bernstein discovered that in a city with population p, the average speed s (in feet per second) that a person walks on main streets can be approximated by the formula

$s (p) = 0.04 + 0.86 \ln p .$ $s (p) = 0.04 + 0.86 \ln p .$
1. What is the average walking speed of pedestrians in Tampa (population 470,000)?
2. What is the average walking speed of pedestrians in Bowman, Georgia (population 450)?
3. What is the estimated population of a town in which the estimated average walking speed is 4.6 feet per second?
Drinking and driving. Just after Eric had his last drink, the alcohol level in his bloodstream was 0.26 (milligram of alcohol per milliliter of blood). After one-half hour, his alcohol level was 0.18. The alcohol level A(t) in a person follows the exponential decay law

$A (t) = A_{0} e^{- k t}, (t in hours),$ $A (t) = A_{0} e^{- k t}, (t in hours),$

where $k > 0$ $k > 0$ depends on the individual.
1. What is the value of $A_{0}$ $A_{0}$ for Eric?
2. What is the value of k for Eric?
3. If the legal driving limit for alcohol level is 0.08, how long should Eric wait (after his last drink) before he will be able to drive legally?
Bacteria culture. The mass m(t), in grams, of a bacteria culture grows according to the logistic growth model

$m (t) = \frac{6}{1 + 5 e^{- 0.7 t}},$ $m (t) = \frac{6}{1 + 5 e^{- 0.7 t}},$

where t is time measured in days.
1. What is the initial mass of the culture?
2. What happens to the mass in the long run?
3. When will the mass reach 5 grams?
A learning model. The number of units n(t) produced per day after t days of training is given by

$n (t) = 60 (1 - e^{- k t}),$ $n (t) = 60 (1 - e^{- k t}),$

where $k > 0$ $k > 0$ is a constant that depends on the individual.
1. Estimate k for Rita, who produced 20 units after one day of training.
2. How many units will Rita produce per day after ten days of training?
3. How many days should Rita train if she is expected to produce 40 units per day?
Music. A jump of one perfect minor third gives a frequency increase of 20%. A perfect minor third is 300 cents. Find the difference in cents. Is the difference noticeable?
Star brightness. The magnitude of star A is 3 more than that of the star B. How is their corresponding brightness related?
Acid rain. In Scotland, the lowest recorded pH level of rainfall was 2.4. What was the concentration of hydrogen ions $[H^{+}]$ $[H^{+}]$ in that rainfall? How much more acidic was this rain than a rain with a pH value of 5.6?
Comparing earthquakes. The 1997 earthquake in Iran registered 7.5 on the Richter scale, and the 2010 earthquake in Chile registered 8.8 on the Richter scale.
1. Compare the intensities of these earthquakes.
2. Compare the energies released by these quakes.
Comparing two sounds. How much more intense is a 115–dB sound than a 95–dB sound?
Comparing intensities. If the intensity of one sound is 5000 times another, what is the difference in the decibel levels of the two sounds?
Sirius-Saturn. How many times brighter is Sirius than Saturn?
Nuclear bomb. What magnitude of earthquake releases energy equivalent to that of a 1 megaton nuclear bomb ( $5 \times 10^{15}$ $5 \times 10^{15}$ joules)?

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Table of Contents for Review Exercises

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Table of Contents for
Review Exercises