Chapter 6 Review Exercises

Cynthia Singleton; Ginny Bradley; Karen Perilloux; Prakash Ghimire

Chapter 6 Exponential and Logarithmic Functions

Chapter 6 Review Exercises

Exponential Functions

Determine whether the function $y = 156 {(0.825)}^{t}$ represents exponential growth, exponential decay, or neither. Explain

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exponential decay; The growth factor, $0.825,$ is between $0$ and $1.$

The population of a herd of deer is represented by the function $A (t) = 205 {(1.13)}^{t},$ where $t$ is given in years. To the nearest whole number, what will the herd population be after $6$ years?

Find an exponential equation that passes through the points $(2, 2 .25)$ and $(5, 60.75) .$

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$y = 0.25 {(3)}^{x}$

Determine whether the following table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

x	1	2	3	4
*f(x)*	3	0.9	0.27	0.081

A retirement account is opened with an initial deposit of $8,500 and earns $8.12$ % interest compounded monthly. What will the account be worth in $20$ years?

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$$ 42, 888.18$

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with $7.5$ % APR, compounded daily, in order to reach her goal in $3$ years?

Does the equation $y = 2.294 e^{- 0.654 t}$ represent continuous growth, continuous decay, or neither? Explain.

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continuous decay; the growth rate is negative.

Suppose an investment account is opened with an initial deposit of $$10,500$ earning $6.25$ % interest, compounded continuously. How much will the account be worth after $25$ years?

Graphs of Exponential Functions

Graph the function $f (x) = 3.5 {(2)}^{x} .$ State the domain and range and give the y-intercept.

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domain: all real numbers; range: all real numbers strictly greater than zero; y-intercept: (0, 3.5);

Graph of f(x)=3.5(2^x)

Graph the function $f (x) = 4 {(\frac{1}{8})}^{x}$ and its reflection about the y-axis on the same axes, and give the y-intercept.

The graph of $f (x) = {6.5}^{x}$ is reflected about the y-axis and stretched vertically by a factor of $7.$ What is the equation of the new function, $g (x) ?$ State its y-intercept, domain, and range.

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$g (x) = 7 {(6.5)}^{- x};$ y-intercept: $(0, 7);$ Domain: all real numbers; Range: all real numbers greater than $0.$

The graph below shows transformations of the graph of $f (x) = 2^{x} .$ What is the equation for the transformation?

Graph of f(x)=2^x

Logarithmic Functions

Rewrite $\log_{17} (4913) = x$ as an equivalent exponential equation.

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$17^{x} = 4913$

Rewrite $\ln (s) = t$ as an equivalent exponential equation.

Rewrite $a^{- \frac{2}{5}} = b$ as an equivalent logarithmic equation.

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$\log_{a} b = - \frac{2}{5}$

Rewrite $e^{- 3.5} = h$ as an equivalent logarithmic equation.

Solve for x if $\log_{64} (x) = \frac{1}{3}$ by converting to exponential form.

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$x = 64^{\frac{1}{3}} = 4$

Evaluate $\log_{5} (\frac{1}{125})$ without using a calculator.

Evaluate $\log (0 .000001)$ without using a calculator.

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$\log (0 .000001) = - 6$

Evaluate $\log (4.005)$ using a calculator. Round to the nearest thousandth.

Evaluate $\ln (e^{- 0.8648})$ without using a calculator.

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$\ln (e^{- 0.8648}) = - 0.8648$

Evaluate $\ln (\sqrt[3]{18})$ using a calculator. Round to the nearest thousandth.

Graphs of Logarithmic Functions

Graph the function $g (x) = \log (7 x + 21) - 4.$

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Graph of g(x)=log(7x+21)-4.

Graph the function $h (x) = 2 \ln (9 - 3 x) + 1.$

State the domain, vertical asymptote, and end behavior of the function $g (x) = \ln (4 x + 20) - 17.$

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Domain: $x > - 5;$ Vertical asymptote: $x = - 5;$ End behavior: as $x \to - 5^{+}, f (x) \to - \infty$ and as $x \to \infty, f (x) \to \infty .$

Logarithmic Properties

Rewrite $\ln (7 r \cdot 11 s t)$ in expanded form.

Rewrite $\log_{8} (x) + \log_{8} (5) + \log_{8} (y) + \log_{8} (13)$ in compact form.

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${log}_{8} (65 x y)$

Rewrite $\log_{m} (\frac{67}{83})$ in expanded form.

Rewrite $\ln (z) - \ln (x) - \ln (y)$ in compact form.

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$\ln (\frac{z}{x y})$

Rewrite $\ln (\frac{1}{x^{5}})$ as a product.

Rewrite $- \log_{y} (\frac{1}{12})$ as a single logarithm.

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${log}_{y} (12)$

Use properties of logarithms to expand $\log (\frac{r^{2} s^{11}}{t^{14}}) .$

Use properties of logarithms to expand $\ln (2 b \sqrt{\frac{b + 1}{b - 1}}) .$

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$\ln (2) + \ln (b) + \frac{\ln (b + 1) - \ln (b - 1)}{2}$

Condense the expression $5 \ln (b) + \ln (c) + \frac{\ln (4 - a)}{2}$ to a single logarithm.

Condense the expression $3 \log_{7} v + 6 \log_{7} w - \frac{\log_{7} u}{3}$ to a single logarithm.

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$\log_{7} (\frac{v^{3} w^{6}}{\sqrt[3]{u}})$

Rewrite $\log_{3} (12.75)$ to base $e .$

Rewrite $5^{12 x - 17} = 125$ as a logarithm. Then apply the change of base formula to solve for $x$ using the common log. Round to the nearest thousandth.

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$x = \frac{\frac{\log (125)}{\log (5)} + 17}{12} = \frac{5}{3}$

Exponential and Logarithmic Equations

Solve $216^{3 x} \cdot 216^{x} = 36^{3 x + 2}$ by rewriting each side with a common base.

Solve $\frac{125}{{(\frac{1}{625})}^{- x - 3}} = 5^{3}$ by rewriting each side with a common base.

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$x = - 3$

Use logarithms to find the exact solution for $7 \cdot 17^{- 9 x} - 7 = 49.$ If there is no solution, write no solution.

Use logarithms to find the exact solution for $3 e^{6 n - 2} + 1 = - 60.$ If there is no solution, write no solution.

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no solution

Find the exact solution for $5 e^{3 x} - 4 = 6$ . If there is no solution, write no solution.

Find the exact solution for $2 e^{5 x - 2} - 9 = - 56.$ If there is no solution, write no solution.

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no solution

Find the exact solution for $5^{2 x - 3} = 7^{x + 1} .$ If there is no solution, write no solution.

Find the exact solution for $e^{2 x} - e^{x} - 110 = 0.$ If there is no solution, write no solution.

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$x = \ln (11)$

Use the definition of a logarithm to solve. $- 5 \log_{7} (10 n) = 5.$

47. Use the definition of a logarithm to find the exact solution for $9 + 6 \ln (a + 3) = 33.$

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$a = e^{4} - 3$

Use the one-to-one property of logarithms to find an exact solution for $\log_{8} (7) + \log_{8} (- 4 x) = \log_{8} (5) .$ If there is no solution, write no solution.

Use the one-to-one property of logarithms to find an exact solution for $\ln (5) + \ln (5 x^{2} - 5) = \ln (56) .$ If there is no solution, write no solution.

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$x = \pm \frac{9}{5}$

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D = 10 \log (\frac{I}{I_{0}}),$ where $I$ is the intensity of the sound in watts per square meter and $I_{0} = 10^{- 12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of $6.3 \cdot 10^{- 3}$ watts per square meter?

The population of a city is modeled by the equation $P (t) = 256, 114 e^{0.25 t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

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about $5.45$ years

Find the inverse function $f^{- 1}$ for the exponential function $f (x) = 2 \cdot e^{x + 1} - 5.$

Find the inverse function $f^{- 1}$ for the logarithmic function $f (x) = 0.25 \cdot \log_{2} (x^{3} + 1) .$

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$f^{- 1} (x) = \sqrt[3]{2^{4 x} - 1}$

Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes $300$ milligrams of a therapeutic drug that decays by about $17$ % each hour.

To the nearest minute, what is the half-life of the drug?

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after $24$ hours. Round to the nearest hundredth of a gram.

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$f (t) = 300 {(0.83)}^{t}; f (24) \approx 3.43 g$

For the following exercises, use this scenario: A soup with an internal temperature of $350°$ Fahrenheit was taken off the stove to cool in a $71°F$ room. After fifteen minutes, the internal temperature of the soup was $175°F .$

Use Newton’s Law of Cooling to write a formula that models this situation.

How many minutes will it take the soup to cool to $85°F?$

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about $45$ minutes

For the following exercises, use this scenario: The equation $N (t) = \frac{1200}{1 + 199 e^{- 0.625 t}}$ models the number of people in a school who have heard a rumor after $t$ days.

How many people started the rumor?

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

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about $8.5$ days

What is the carrying capacity?

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

x	f(x)
1	3.05
2	4.42
3	6.4
4	9.28
5	13.46
6	19.52
7	28.3
8	41.04
9	50.5
10	86.28

Show Solution

exponential

Graph of the table’s values.

x	*f(x)*
0.5	18.05
1	17
3	15.33
5	14.55
7	14.04
10	13.5
12	13.22
13	13.1
15	12.88
17	12.69
20	12.45

Find a formula for an exponential equation that goes through the points $(- 2, 100)$ and $(0, 4) .$ Then express the formula as an equivalent equation with base e.

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$y = 4 {(0.2)}^{x};$ $y = 4 e^{-1 .609438 x}$

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Exponential Functions

Graphs of Exponential Functions

Logarithmic Functions

Graphs of Logarithmic Functions

Logarithmic Properties

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

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