Exponential and Logarithmic Functions

Exponential and Logarithmic Functions
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0.6

1. Starting from a single cell, a human being is formed by 50 generations of cell division. Explain why after n divisions there are 2ⁿ cells. Guess how many cells will be present after 50 divisions, then compute 2⁵⁰. Briefly discuss how rapidly exponential functions increase.

2. Explain why the graphs of f (x) = 2^{- x} and are the same.

3. Compare f (x) = x² and g(x) = 2^x for , x = 1 , x = 2 , x = 3, and x = 4. In general, which function is bigger for large values of x? For small values of x?

4. Compare f (x) = 2^x and g(x) = 3^x for x = -2, , and x = 2. In general, which function is bigger for negative values of x? For positive values of x?

In exercises 5-12, convert each exponential expression into fractional or root form.

5. 2^{- 3} 6. 4^{- 2}

7. 3^1/2 8. 7^1/3

9. 5^2/3 10. 6^2/5

11. 4^{- 2/3} 12. 6^{- 1/3}

In exercises 13-20, convert each expression into exponential form.

13. 14.

15. 16.

17. 18.

19. 20.

In exercises 21-24, find the integer value of the given expression (no calculators!).

21. 4^3/2 22. 8^2/3

23. 24.

In exercises 25-32, use a calculator or computer to estimate each value.

25. e² 26. 3e

27. 2e^{- 1/2} 28. 4e^{- 2/3}

29. 30.

31. 10e^-0.01 32. 10e^-0.001

In exercises 33-50, sketch a graph of the given function.

33. f (x) = e^2x 34. f (x) = e^3x

35. f (x) = 2e^x/4 36. f (x) = 3e^x/6

37. f (x) = 3e^{- 2x} 38. f (x) = 10e^{- x/3}

39. 40.

41. f (x) = xe^{- 2x} 42. f (x) = x²e^{- 2x}

43. f (x) = ln 2x 44. f (x) = ln 3x

45. f (x) = ln x² 46. f (x) = ln x³

47. f (x) = e^{2ln x} 48. f (x) = e^{- ln x}

49. f (x) = e^{- x/4}sin x 50. f (x) = e^{- x/6}cos x

In exercises 51-60, solve the given equation for x.

51. e^2x = 2 52. e^4x = 3

53. 2e^{- 2x} = 1 54. 3e^{- x/2} = 2

55. ln 2x = 4 56. 2ln 3x = 1

57. 2ln 4x -1 = 6 58. 3ln (2 - x) = 6

59. e^{2ln x} = 4 60. ln (e^2x) = 6

In exercises 61 and 62, use the definition of logarithm to determine the value.

61. (a) log ₃9 (b) log ₄64 (c) log ₃ 62. (a) log ₄ (b) log ₄2 (c) log ₉3

In exercises 63 and 64, use equation (6.5) to approximate the value.

63. (a) log ₃7 (b) log ₄60 (c) log ₃ 64. (a) log ₄ (b) log ₄3 (c) log ₉8

In exercises 65-70, rewrite the expression as a single logarithm.

65. ln 3 - ln 4 66. 2ln 4 - ln 3

67. ln 4 - ln 2 68. 3ln 2 - ln

69. ln + 4ln 2 70. ln 9-2ln 3

In exercises 71-76, find a function of the form f (x) with the given function values.

71. f (0) = 2, f (2) = 6 72. f (0) = 3, f (3) = 4

73. f (0) = 4, f (2) = 2 74. f (0) = 5, f (1) = 2

75. f (0) = -2, f (1) = - 3 76. f (1) = 2, f (2) = 4

77. A fast-food restaurant gives every customer a game ticket. With each ticket, the customer has a 1 - in -10 chance of winning a free meal. If you go 10 times, estimate your chances of winning at least one free meal. The exact probability is . Compute this number and compare it to your guess.

78. In exercise 77, if you had 20 tickets with a 1 - in -20 chance of winning, would you expect your probability of winning to increase or decrease? Compute the probability to find out.

79. In general, if you have n chances of winning with a 1 - in - n chance on each try, the probability of winning at least once is . As n gets larger, what number does this probability approach? (Hint: There is a very good reason that this question is in this section!)

80. If y = a x^m , show that ln y = ln a + mln x. If v = ln y , u = ln x and b = ln a , show that v = mu + b. Explain why the graph of v as a function of u would be a straight line. This graph is called the log-log plot of y and x.

81. For the given data, compute v = ln y and u = ln x and plot points (u, v). Find constants m and b such that v = mu + b and use the results of exercise 80 to find a constant a such that y = a x^m.

x 2.2 2.4 2.6 2.8 3.0 3.2

y 14.52 17.28 20.28 23.52 27.0 30.72

82. Repeat exercise 81 for the given data.

x 2.8 3.0 3.2 3.4 3.6 3.8

y 9.37 10.39 11.45 12.54 13.66 14.81

83. Construct a log-log plot (see exercise 80) of the U. S. population data in Example 6.14. Compared to the semi-log plot of the data in Figure 0.60, does the log-log plot look linear? Based on this, is the population data modeled better by an exponential function or a polynomial (power) function?

84. Construct a semi-log plot of the data in exercise 81. Compared to the log-log plot already constructed, does this plot look linear? Based on this, is this data better modeled by an exponential or power function?

85. The concentration [H⁺] of free hydrogen ions in a chemical solution determines the solution's pH, as defined by pH = log [H⁺]. Find [H⁺] if the pH equals (a) 7, (b) 8, and (c) 9. For each increase in pH of 1, by what factor does [H⁺] change?

86. Gastric juice is considered an acid, with a pH of about 2.5. Blood is considered alkaline, with a pH of about 7.5. Compare the concentrations of hydrogen ions in the two substances (see exercise 85).

87. The Richter magnitude M of an earthquake is defined in terms of the energy E in joules released by the earthquake, with log ₁₀E = 4.4 + 1.5M. Find the energy for earthquakes with magnitudes (a) 4, (b) 5, and (c) 6. For each increase in M of one, by what factor does E change?

88. It puzzles some people who have not grown up around earthquakes that a magnitude 6 quake is considered much more severe than a magnitude 3 quake. Compare the amount of energy released in the two quakes.

89. The decibel level of a noise is defined in terms of the intensity I of the noise, with dB = 10log ₁₀(I/I₀). Here, I₀ = is the intensity of a barely audible sound. Compute the intensity levels of sounds with (a) dB = 80 , (b) dB = 90, and (c) dB = 100. For each increase in decibels of 10, by what factor does I change?

90. At a basketball game, a courtside decibel meter shows crowd noises ranging from 60 dB to 110 dB. How much louder is the 110 - dB noise compared to the 60 - dB noise (see exercise 89)?

91. Graph y = xe^{- x} , y = xe^{- 2x} , y = xe^{- 3x} and so on. Estimate the locations of the maximum for each. In general, state a rule for the location of the maximum of y = xe^{- kx}.

92. In golf, the task is to hit a golf ball into a small hole. If the ground near the hole is not flat, the golfer must judge how much the ball will curve. Suppose the golfer is at the point ( -13,0) , the hole is at the point (0, 0), and the path of the ball is, for - 13 x 0 , y = -1.672x + 72ln (1 + 0.02x). Show that the ball goes in the hole and estimate the point on the y - axis at which the golfer aimed.

93. On whichever calculators and/or computers you have access to, try to evaluate ( -8)^2/3. If you do not get 4, try computing [ ( -8)²] ^1/3. Show that, theoretically, ( -8)^2/3 = . To understand why a calculator might report a difference in the two quantities, explain why ( -8)^2/4 [ ( -8)²] ^1/4. Investigate why some machines report that ( -8)^2/3 does not exist.

94. Graph y = x² and y = 2^x and approximate the two positive solutions of the equation x² = 2^x. Graph y = x³ and y = 3^x and approximate the two positive solutions of the equation x³ = 3^x. Explain why x = a will always be a solution of x^a = a^x , a > 0. What is different about the role of x = 2 as a solution of x² = 2^x compared to the role of x = 3 as a solution of x³ = 3^x ? To determine the a - value at which the change occurs, graphically solve x^a = a^x for a = 2.1, 2.2, , 2.9 and note that a = 2.7 and a = 2.8 behave differently. Continue to narrow down the interval of change by testing a = 2.71, 2.72, , 2.79. Then guess the exact value of a.

95. Graph y = ln x and describe the behavior near x = 0. Then graph y = x ln x and describe the behavior near x = 0. Repeat this for y = x²ln x, y = x^1/2 ln x , and y = x^a ln x for a variety of positive constants a. Because the function “blows up” at x = 0 , we say that y = ln x has a singularity at x = 0. The order of the singularity at x = 0 of a function f (x) is the smallest value of a such that y = x^a f (x) doesn't have a singularity at x = 0. Determine the order of the singularity at x = 0 for (a) (b) and (c) The higher the order of the singularity, the “worse” the singularity is. Based on your work, how bad is the singularity of y = ln x at x = 0 ?