Jay Abramson

Review Exercises

Exponential Functions

1.

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

2.

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?

3.

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

4.

Determine whether Table 1 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

Table 1

5.

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?

6.

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?

7.

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

8.

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

9.

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

10.

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.

11.

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.

12.

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

Figure 1

Logarithmic Functions

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

Evaluate $\log (0.000001)$ without using a calculator.

20.

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

21.

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

22.

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

Graphs of Logarithmic Functions

23.

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

24.

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

25.

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

Logarithmic Properties

26.

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

27.

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

28.

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

29.

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

30.

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

31.

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

32.

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

33.

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

34.

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

35.

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

36.

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

37.

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.

Exponential and Logarithmic Equations

38.

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

39.

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

40.

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

41.

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

42.

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

43.

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

44.

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

45.

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

46.

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 $9 + 6 \ln (a + 3) = 33.$

48.

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.

49.

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.

50.

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?

51.

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?

52.

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

53.

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

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.

54.

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

55.

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.

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 $175°F .$

56.

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

57.

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

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.

58.

How many people started the rumor?

59.

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

60.

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.

61.

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	59.5
10	86.28

62.

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

63.

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.

Fitting Exponential Models to Data

64.

What is the carrying capacity for a population modeled by the logistic equation $P (t) = \frac{250, 000}{1 + 499 e^{- 0.45 t}} ?$ What is the initial population for the model?

65.

The population of a culture of bacteria is modeled by the logistic equation $P (t) = \frac{14, 250}{1 + 29 e^{- 0.62 t}},$ where $t$ is in days. To the nearest tenth, how many days will it take the culture to reach $75 %$ of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

66.

x	*f(x)*
1	409.4
2	260.7
3	170.4
4	110.6
5	74
6	44.7
7	32.4
8	19.5
9	12.7
10	8.1

67.

x	*f(x)*
0.15	36.21
0.25	28.88
0.5	24.39
0.75	18.28
1	16.5
1.5	12.99
2	9.91
2.25	8.57
2.75	7.23
3	5.99
3.5	4.81

68.

x	*f(x)*
0	9
2	22.6
4	44.2
5	62.1
7	96.9
8	113.4
10	133.4
11	137.6
15	148.4
17	149.3