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Review Exercises

Exponential Functions

1.

Determine whether the function y=156 ( 0.825 ) t 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 , A(t)=205 (1.13) t , where t t is given in years. To the nearest whole number, what will the herd population be after 6 6 years?

3.

Find an exponential equation that passes through the points (2, 2.25) (2, 2.25) and (5,60.75). (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% 8.12% interest compounded monthly. What will the account be worth in 20 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% 7.5% APR, compounded daily, in order to reach her goal in 3 3 years?

7.

Does the equation y=2.294 e 0.654t y=2.294 e 0.654t represent continuous growth, continuous decay, or neither? Explain.

8.

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

Graphs of Exponential Functions

9.

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

10.

Graph the function f(x)=4 ( 1 8 ) x f(x)=4 ( 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 f(x)= 6.5 x is reflected about the y-axis and stretched vertically by a factor of 7. 7. What is the equation of the new function, g(x)? g(x)? State its y-intercept, domain, and range.

12.

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

Graph of f(x)=2^x
Figure 1

Logarithmic Functions

13.

Rewrite log 17 ( 4913 )=x log 17 ( 4913 )=x as an equivalent exponential equation.

14.

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

15.

Rewrite a 2 5 =b a 2 5 =b as an equivalent logarithmic equation.

16.

Rewrite e 3.5 =h e 3.5 =h as an equivalent logarithmic equation.

17.

Solve for x if log 64 (x)= 1 3 log 64 (x)= 1 3 by converting the logarithmic equation log 64 (x)= 1 3 log 64 (x)= 1 3 to exponential form.

18.

Evaluate log 5 ( 1 125 ) log 5 ( 1 125 ) without using a calculator.

19.

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

20.

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

21.

Evaluate ln( e 0.8648 ) ln( e 0.8648 ) without using a calculator.

22.

Evaluate ln( 18 3 ) ln( 18 3 ) using a calculator. Round to the nearest thousandth.

Graphs of Logarithmic Functions

23.

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

24.

Graph the function h(x)=2ln( 93x )+1. h(x)=2ln( 93x )+1.

25.

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

Logarithmic Properties

26.

Rewrite ln( 7r11st ) ln( 7r11st ) in expanded form.

27.

Rewrite log 8 ( x )+ log 8 ( 5 )+ log 8 ( y )+ log 8 ( 13 ) log 8 ( x )+ log 8 ( 5 )+ log 8 ( y )+ log 8 ( 13 ) in compact form.

28.

Rewrite log m ( 67 83 ) log m ( 67 83 ) in expanded form.

29.

Rewrite ln( z )ln( x )ln( y ) ln( z )ln( x )ln( y ) in compact form.

30.

Rewrite ln( 1 x 5 ) ln( 1 x 5 ) as a product.

31.

Rewrite log y ( 1 12 ) log y ( 1 12 ) as a single logarithm.

32.

Use properties of logarithms to expand log( r 2 s 11 t 14 ). log( r 2 s 11 t 14 ).

33.

Use properties of logarithms to expand ln( 2b b+1 b1 ). ln( 2b b+1 b1 ).

34.

Condense the expression 5ln( b )+ln( c )+ ln( 4a ) 2 5ln( b )+ln( c )+ ln( 4a ) 2 to a single logarithm.

35.

Condense the expression 3 log 7 v+6 log 7 w log 7 u 3 3 log 7 v+6 log 7 w log 7 u 3 to a single logarithm.

36.

Rewrite log 3 ( 12.75 ) log 3 ( 12.75 ) to base e. e.

37.

Rewrite 5 12x17 =125 5 12x17 =125 as a logarithm. Then apply the change of base formula to solve for x x using the common log. Round to the nearest thousandth.

Exponential and Logarithmic Equations

38.

Solve 216 3x 216 x = 36 3x+2 216 3x 216 x = 36 3x+2 by rewriting each side with a common base.

39.

Solve 125 ( 1 625 ) x3 = 5 3 125 ( 1 625 ) x3 = 5 3 by rewriting each side with a common base.

40.

Use logarithms to find the exact solution for 7 17 9x 7=49. 7 17 9x 7=49. If there is no solution, write no solution.

41.

Use logarithms to find the exact solution for 3 e 6n2 +1=60. 3 e 6n2 +1=60. If there is no solution, write no solution.

42.

Find the exact solution for 5 e 3x 4=6 5 e 3x 4=6 . If there is no solution, write no solution.

43.

Find the exact solution for 2 e 5x2 9=56. 2 e 5x2 9=56. If there is no solution, write no solution.

44.

Find the exact solution for 5 2x3 = 7 x+1 . 5 2x3 = 7 x+1 . If there is no solution, write no solution.

45.

Find the exact solution for e 2x e x 110=0. e 2x e x 110=0. If there is no solution, write no solution.

46.

Use the definition of a logarithm to solve. 5 log 7 ( 10n )=5. 5 log 7 ( 10n )=5.

47.

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

48.

Use the one-to-one property of logarithms to find an exact solution for log 8 ( 7 )+ log 8 ( 4x )= log 8 ( 5 ). log 8 ( 7 )+ log 8 ( 4x )= 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 ). 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 D is defined by the equation D=10log( I I 0 ), D=10log( I I 0 ), where I I is the intensity of the sound in watts per square meter and I 0 = 10 12 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 10 3 6.3 10 3 watts per square meter?

51.

The population of a city is modeled by the equation P(t)=256,114 e 0.25t P(t)=256,114 e 0.25t where t 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 f 1 for the exponential function f( x )=2 e x+1 5. f( x )=2 e x+1 5.

53.

Find the inverse function f 1 f 1 for the logarithmic function f( x )=0.25 log 2 ( x 3 +1 ). f( x )=0.25 log 2 ( x 3 +1 ).

Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes 300 300 milligrams of a therapeutic drug that decays by about 17% 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 t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 24 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° 350° Fahrenheit was taken off the stove to cool in a 71°F 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F. 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? 85°F?

For the following exercises, use this scenario: The equation N( t )= 1200 1+199 e 0.625t N( t )= 1200 1+199 e 0.625t models the number of people in a school who have heard a rumor after t 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.
xf(x)
13.05
24.42
36.4
49.28
513.46
619.52
728.3
841.04
959.5
1086.28
62.
xf(x)
0.518.05
117
315.33
514.55
714.04
1013.5
1213.22
1313.1
1512.88
1712.69
2012.45
63.

Find a formula for an exponential equation that goes through the points ( 2,100 ) ( 2,100 ) and ( 0,4 ). ( 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)= 250,000 1+499 e 0.45t ? P(t)= 250,000 1+499 e 0.45t ? What is the initial population for the model?

65.

The population of a culture of bacteria is modeled by the logistic equation P(t)= 14,250 1+29 e 0.62t , P(t)= 14,250 1+29 e 0.62t , where t t is in days. To the nearest tenth, how many days will it take the culture to reach 75% 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.
xf(x)
1409.4
2260.7
3170.4
4110.6
574
644.7
732.4
819.5
912.7
108.1
67.
xf(x)
0.1536.21
0.2528.88
0.524.39
0.7518.28
116.5
1.512.99
29.91
2.258.57
2.757.23
35.99
3.54.81
68.
xf(x)
09
222.6
444.2
562.1
796.9
8113.4
10133.4
11137.6
15148.4
17149.3
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