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3. Lesson 10-1 Exponential Functions
Lesson 10-2 Logarithms and Logarithmic Functions
Lesson 10-3 Properties of Logarithms
Lesson 10-4 Common Logarithms
Lesson 10-5 Base e and Natural Logarithms
Lesson 10-6 Exponential Growth and Decay
Contents

4. Example 1 Graph an Exponential Function
Example 2 Identify Exponential Growth and Decay
Example 3 Write an Exponential Function
Example 4 Simplify Expressions with Irrational Exponents
Example 5 Solve Exponential Equations
Example 6 Solve Exponential Inequalities
Lesson 1 Contents

5. x Sketch the graph of . Then state the function’s domain and range.
Make a table of values. Connect the points to sketch a smooth curve. 16 2 4 1 –1 –2 x
Example 1-1a

6. Answer:
The domain is all real numbers, while the range is all positive numbers.
Example 1-1b

7. Sketch the graph of Then state the function’s domain and range.
Answer:
The domain is all real numbers; the range is all positive numbers.
Example 1-1c

8. Determine whether represents exponential growth or decay.
Answer: The function represents exponential decay, since the base, 0.7, is between 0 and 1.
Example 1-2a

9. Determine whether represents exponential growth or decay.
Answer: The function represents exponential growth, since the base, 3, is greater than 1.
Example 1-2b

10. Determine whether represents exponential growth or decay.
Answer: The function represents exponential growth,
since the base, is greater than 1.
Example 1-2c

11. Answer: The function represents exponential decay, since the
Determine whether each function represents exponential growth or decay. a. b. c.
Answer: The function represents exponential decay, since the base, 0.5, is between 0 and 1.
Answer: The function represents exponential growth, since the base, 2, is greater than 1.
Answer: The function represents exponential decay, since the
base, is between 0 and 1.
Example 1-2d

12. Cellular Phones In December of 1990, there were 5,283,000 cellular telephone subscribers in the United States. By December of 2000, this number had risen to 109,478,000.
Write an exponential function of the form that could be used to model the number of cellular telephone subscribers y in the U.S. Write the function in terms of x, the number of years since 1990.
For 1990, the time x equals 0, and the initial number of cellular telephone subscribers y is 5,283,000. Thus the y-intercept, and the value of a, is 5,283,000.
For 2000, the time x equals 2000 – 1990 or 10, and the number of cellular telephone subscribers is 109,478,000.
Example 1-3a

13. Replace x with 10, y with 109,478,000 and a with 5,283,000.
Substitute these values and the value of a into an exponential function to approximate the value of b.
Exponential function
Replace x with 10, y with 109,478,000 and a with 5,283,000.
Divide each side by 5,283,000.
Take the 10th root of each side.
Example 1-3b

14. To find the 10th root of 20. 72, use selection
To find the 10th root of 20.72, use selection under the MATH menu on the TI-83 Plus.
ENTER
MATH
Keystrokes: 10
Answer: An equation that models the number of cellular telephone subscribers in the U.S. from 1990 to is
Example 1-3c

15. For 2010, the time x equals 2010 – 1990 or 20.
Suppose the number of telephone subscribers continues to increase at the same rate. Estimate the number of US subscribers in 2010.
For 2010, the time x equals 2010 – 1990 or 20.
Modeling equation
Replace x with 20.
Use a calculator.
Answer: The number of cell phone subscribers will be about 2,136,000,000 in 2010.
Example 1-3d

16. Health In 1991, 4. 9% of Americans had diabetes
Health In 1991, 4.9% of Americans had diabetes. By 2000, this percent had risen to 7.3%.
a. Write an exponential function of the form could be used to model the percentage of Americans with diabetes. Write the function in terms of x, the number of years since 1991.
b. Suppose the percent of Americans with diabetes continues to increase at the same rate. Estimate the percent of Americans with diabetes in 2010.
Answer:
Answer: 11.4%
Example 1-3e

17. Simplify .
Quotient of Powers
Answer:
Example 1-4a

18. Simplify .
Power of a Power
Product of Radicals
Answer:
Example 1-4b

19. Simplify each expression. a.b.
Answer:
Answer:
Example 1-4c

20. Rewrite 256 as 44 so each side has the same base.
Solve .
Original equation
Rewrite 256 as 44 so each side has the same base.
Property of Equality for Exponential Functions
Add 2 to each side.
Divide each side by 9.
Answer: The solution is
Example 1-5a

21. Check Original equation Substitute for n. Simplify. Simplify.
Example 1-5b

22. Rewrite 9 as 32 so each side has the same base.
Solve .
Original equation
Rewrite 9 as 32 so each side has the same base.
Property of Equality for Exponential Functions
Distributive Property
Subtract 4x from each side.
Answer: The solution is
Example 1-5c

23. Solve each equation. a. b.
Answer:
Answer: 1
Example 1-5d

24. Property of Inequality for Exponential Functions
Solve
Original inequality
Rewrite as
Property of Inequality for Exponential Functions
Subtract 3 from each side.
Divide each side by –2.
Answer: The solution is
Example 1-6a

25. Check: Test a value of k less than for example,
Original inequality
Replace k with 0.
Simplify.
Example 1-6b

26. Solve
Answer:
Example 1-6c

27. End of Lesson 1

28. Example 1 Logarithmic to Exponential Form
Example 2 Exponential to Logarithmic Form
Example 3 Evaluate Logarithmic Expressions
Example 4 Inverse Property of Exponents and Logarithms
Example 5 Solve a Logarithmic Equation
Example 6 Solve a Logarithmic Inequality
Example 7 Solve Equations with Logarithms on Each Side
Example 8 Solve Inequalities with Logarithms on Each Side
Lesson 2 Contents

29. Write in exponential form.
Answer:
Example 2-1a

30. Write in exponential form.
Answer:
Example 2-1b

31. Write each equation in exponential form. a.b.
Answer:
Answer:
Example 2-1c

32. Write in logarithmic form.
Answer:
Example 2-2a

33. Write in logarithmic form.
Answer:
Example 2-2b

34. Write each equation in logarithmic form. a.b.
Answer:
Answer:
Example 2-2c

35. Let the logarithm equal y.
Evaluate
Let the logarithm equal y.
Definition of logarithm
Property of Equality for Exponential Functions
Answer: So,
Example 2-3a

36. Evaluate
Answer: 3
Example 2-3b

37. Evaluate .
Answer:
Example 2-4a

38. Evaluate .
Answer:
Example 2-4b

39. Evaluate each expression. a.b.
Answer: 3
Answer:
Example 2-4c

40. Definition of logarithm
Solve
Original equation
Definition of logarithm
Power of a Power
Simplify.
Answer:
Example 2-5a

41. Solve
Answer: 9
Example 2-5b

42. Solve Check your solution.
Original inequality
Logarithmic to exponential inequality
Simplify.
Answer: The solution set is
Example 2-6a

43. Check Try 64 to see if it satisfies the inequality.
Original inequality
Substitute 64 for x.
Example 2-6b

44. Solve Check your solution.
Answer:
Example 2-6c

45. Solve Check your solution.
Original equation
Property of Equality for Logarithmic Functions
Subtract 4x and add 3 to each side.
Factor.
Zero Product Property or
Solve each equation.
Example 2-7a

46. Check Substitute each value into the original equation.
Substitute 3 for x.
Simplify.
Original equation
Substitute 1 for x.
Simplify.
Answer: The solutions are 3 and 1.
Example 2-7b

47. Solve Check your solution.
Answer: The solutions are 3 and –2.
Example 2-7c

48. Property of Inequality for Logarithmic Functions
Solve
Original inequality
Property of Inequality for Logarithmic Functions
Addition and Subtraction Properties of Inequalities
We must exclude all values of x such that or Thus the solution set is and This compound inequality simplifies to
Answer: The solution set is
Example 2-8a

49. Solve
Answer:
Example 2-8b

50. End of Lesson 2

51. Example 1 Use the Product Property Example 2 Use the Quotient Property
Example 3 Use Properties of Logarithms
Example 4 Power Property of Logarithms
Example 5 Solve Equations Using Properties of Logarithms
Lesson 3 Contents

52. Use to approximate the value of
Replace 250 with 53 • 2.
Product Property
Inverse Property of Exponents and Logarithms
Replace with
Answer: Thus, is approximately
Example 3-1a

53. Use to approximate the value of
Answer:
Example 3-1b

54. Use and to approximate the value of
Replace 4 with the
quotient
Quotient Property
and
Answer: Thus is approximately
Example 3-2a

55. Use and to approximate the value of
Answer:
Example 3-2b

56. Let L1 be the loudness of one lawnmower running.
Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by a lawnmower has a relative intensity of 109 or 90 decibels. Would the sound of ten lawnmowers running at that same intensity be ten times as loud or 900 decibels? Explain your reasoning.
Let L1 be the loudness of one lawnmower running.
Let L2 be the loudness of ten lawnmowers running.
Example 3-3a

57. Then the increase in loudness is L2 – L1.
Substitute for L1 and L2.
Product Property
Distributive Property
Subtract.
Inverse Property of Exponents and Logarithms
Example 3-3b

58. Answer:. No; the sound of ten lawnmowers is perceived
Answer: No; the sound of ten lawnmowers is perceived to be only 10 decibels as loud as the sound of one lawnmower, or 100 decibels.
Example 3-3c

59. Sound The loudness L of a sound in decibels is given by
Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by fireworks has a relative intensity of 1014 or 140 decibels. Would the sound of ten fireworks of that same intensity be ten times as loud or 1400 decibels? Explain your reasoning.
Answer: No; the sound of ten fireworks is perceived to be only 10 more decibels as loud as the sound of one firework, or 150 decibels.
Example 3-3d

60. Given that approximate the value of
Replace 216 with 63.
Power Property
Replace with
Answer:
Example 3-4a

61. Given that approximate the value of
Answer:
Example 3-4b

62. Property of Equality for Logarithmic Functions
Solve .
Original equation
Power Property
Quotient Property
Property of Equality for Logarithmic Functions
Multiply each side by 5.
Example 3-5a

63. Take the 4th root of each side.
Answer:
Take the 4th root of each side.
Example 3-5b

64. Definition of logarithm
Solve .
Original equation
Product Property
Definition of logarithm
Subtract 64 from each side.
Factor.
Zero Product Property or
Solve each equation.
Example 3-5c

65. Check Substitute each value into the original equation.
Replace x with –4.
Since log8 (–4) and log8 (–16) are undefined, –4 is an extraneous solution and must be eliminated.
Replace x with 16.
Product Property
Example 3-5d

66. Definition of logarithm
Answer: The only solution is
Example 3-5d

67. Solve each equation. a. b.
Answer: 12
Answer: 8
Example 3-5e

68. End of Lesson 3

69. Example 1 Find Common Logarithms
Example 2 Solve Logarithmic Equations Using Exponentiation
Example 3 Solve Exponential Equations Using Logarithms
Example 4 Solve Exponential Inequalities Using Logarithms
Example 5 Change of Base Formula
Lesson 4 Contents

70. Use a calculator to evaluate log 6 to four decimal places.
ENTER
LOG
Keystrokes: 6
Answer: about
Example 4-1a

71. Use a calculator to evaluate log 0.35 to four decimal places.
ENTER
LOG
Keystrokes:
0.35 –
Answer: about –0.4559
Example 4-1b

72. Use a calculator to evaluate each expression to four decimal places.
a. log 5
b. log 0.62
Answer:
Answer: –0.2076
Example 4-1c

73. Write each side using 10 as a base.
Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log The San Fernando Valley earthquake of 1994 measured 6.6 on the Richter scale. How much energy did this earthquake release?
Write the formula.
Replace M with 6.6.
Simplify.
Write each side using 10 as a base.
Example 4-2a

74. Inverse Property of Exponents and Logarithms
Use a calculator.
Answer: The amount of energy released was about ergs.
Example 4-2b

75. Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log In 1999 an earthquake in Turkey measured 7.4 on the Richter scale. How much energy did this earthquake release?
Answer: about
Example 4-2c

76. Property of Equality for Logarithmic Functions
Solve
Original equation
Property of Equality for Logarithmic Functions
Power Property of Logarithms
Divide each side by log 62.
Use a calculator.
Answer:
Example 4-3a

77. Check You can check this answer by using a calculator or by using estimation. Since and the value of x is between 2 and 3. Thus, is a reasonable solution.
Example 4-3b

78. Solve
Answer:
Example 4-3c

79. Solve Original inequality
Property of Inequality for Logarithmic Functions
Power Property of Logarithms
Distributive Property
Subtract 5x log 3 from each side.
Example 4-4a

80. Use a calculator. Simplify. Distributive Property Divide each side by
Switch > to < because is negative.
Use a calculator.
Simplify.
Example 4-4b

81. Negative Exponent Property
Check:
Original inequality
Replace x with 0.
Simplify.
Negative Exponent Property
Answer: The solution set is
Example 4-4d

82. Solve
Answer:
Example 4-4e

83. Answer: The value of is approximately 2.6309.
Express in terms of common logarithms. Then approximate its value to four decimal places.
Change of Base Formula
Use a calculator.
Answer: The value of is approximately
Example 4-5a

84. Express. in terms of common logarithms
Express in terms of common logarithms. Then approximate its value to four decimal places.
Answer:
Example 4-5b

85. End of Lesson 4

86. Example 1 Evaluate Natural Base Expressions
Example 2 Evaluate Natural Logarithmic Expressions
Example 3 Write Equivalent Expressions
Example 4 Inverse Property of Base e and Natural Logarithms
Example 5 Solve Base e Equations
Example 6 Solve Base e Inequalities
Example 7 Solve Natural Log Equations and Inequalities
Lesson 5 Contents

87. Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] 0.5
Answer: about
Example 5-1a

88. Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] –8
Answer: about
Example 5-1b

89. Use a calculator to evaluate each expression to four decimal places. b.
Answer:
Answer:
Example 5-1c

90. Use a calculator to evaluate In 3 to four decimal places.
Keystrokes:
ENTER LN 3
Answer: about
Example 5-2d

91. Use a calculator to evaluate In to four decimal places.
Keystrokes:
ENTER LN
1 ÷ 4 –
Answer: about –1.3863
Example 5-2e

92. Use a calculator to evaluate each expression to four decimal places.
a. In 2
b. In
Answer:
Answer: –0.6931
Example 5-2f

93. Write an equivalent logarithmic equation for .
Answer:
Example 5-3a

94. Write an equivalent exponential equation for
Answer:
Example 5-3b

95. Write an equivalent exponential or logarithmic equation. a.b.
Answer:
Answer:
Example 5-3c

96. Evaluate
Answer:
Example 5-4a

97. Evaluate .
Answer:
Example 5-4b

98. Evaluate each expression. a.b.
Answer: 7
Answer:
Example 5-4c

99. Subtract 4 from each side.
Solve
Original equation
Subtract 4 from each side.
Divide each side by 3.
Property of Equality for Logarithms
Inverse Property of Exponents and Logarithms
Divide each side by –2.
Use a calculator.
Answer: The solution is about –
Example 5-5a

100. Check You can check this value by substituting –0
Check You can check this value by substituting – into the original equation or by finding the intersection of the graphs of and
Example 5-5b

101. Solve
Answer:
Example 5-5c

102. What is the balance after 8 years?
Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
What is the balance after 8 years?
Continuous compounding formula
Replace P with 700, r with 0.06, and t with 8.
Simplify.
Use a calculator.
Answer: The balance after 8 years would be $
Example 5-6a

103. How long will it take for the balance in your account to reach at least $2000?
The balance is at least $2000.
2000
Replace A with 700e(0.06)t. A 
Write an inequality.
Divide each side by 700.
Property of Inequality for Logarithms
Inverse Property of Exponents and Logarithms
Example 5-6b

104. Divide each side by 0.06. Use a calculator.
Answer: It will take at least 17.5 years for the balance to reach $2000.
Example 5-6c

105. a. What is the balance after 7 years?
Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
a. What is the balance after 7 years?
b. How long will it take for the balance in your account to reach at least $2500?
Answer: $
Answer: at least years
Example 5-6d

106. Write each side using exponents and base e.
Solve
Original equation
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Divide each side by 3.
Use a calculator.
Answer: The solution is Check this solution using substitution or graphing.
Example 5-7a

107. Write each side using exponents and base e.
Solve
Original inequality
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Add 3 to each.
Divide each side by 2.
Use a calculator.
Example 5-7b

108. Answer: The solution is all numbers less than. 7
Answer: The solution is all numbers less than and greater than 1.5. Check this solution using substitution.
Example 5-7c

109. Solve each equation or inequality. a.b.
Answer: about
Answer:
Example 5-7d

110. End of Lesson 5

111. Example 1 Exponential Decay of the Form y = a(1 – r)t
Example 2 Exponential Decay of the Form y = ae–kt
Example 3 Exponential Growth of the Form y = a(1 + r )t
Example 4 Exponential Growth of the Form y = aekt
Lesson 6 Contents

112. Caffeine A cup of coffee contains 130 milligrams of caffeine
Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 90% of this caffeine to be eliminated from a person’s body?
Explore
The problem gives the amount of caffeine consumed and the rate at which the caffeine is eliminated. It asks you to find the time it will take for 90% of the caffeine to be eliminated from a person’s body.
Use the formula Let t be the number of hours since drinking the coffee. The amount remaining y is 10% of 130 or 13.
Plan
Example 6-1a

113. Exponential decay formula
Solve
Exponential decay formula
Replace y with 13, a with 130, and r with 0.11.
Divide each side by 130.
Property of Equality for Logarithms
Power Property for Logarithms
Divide each side by log 0.89.
Use a calculator.
Example 6-1b

114. Exponential decay formula
Answer: It will take approximately 20 hours for 90% of the caffeine to be eliminated from a person’s body.
Examine
Use the formula to find how much of the original 130 milligrams of caffeine would remain after 20 hours.
Exponential decay formula
Replace a with 130, r with and t with 20.
Ten percent of 130 is 13, so the answer seems reasonable.
Example 6-1c

115. Caffeine A cup of coffee contains 130 milligrams of caffeine
Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 80% of this caffeine to be eliminated from a person’s body?
Answer: hours
Example 6-1d

116. Exponential decay formula
Geology The half-life of Sodium-22 is 2.6 years. What is the value of k for Sodium-22?
Exponential decay formula
Replace y with 0.5a and t with 2.6.
Divide each side by a.
Property of Equality for Logarithmic Functions
Inverse Property of Exponents and Logarithms
Divide each side by –2.6.
Example 6-2a

117. Use a calculator.
Answer: The constant k for Sodium-22 is Thus, the equation for the decay of Sodium-22 is where t is given in years.
Example 6-2b

118. Formula for the decay of Sodium-22
A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth?
Formula for the decay of Sodium-22
Replace y with 0.1a.
Divide each side by a.
Property of Equality for Logarithms
Example 6-2c

119. Inverse Property for Exponents and Logarithms
Divide each side by –
Use a calculator.
Answer: It was formed about 9 years ago.
Example 6-2d

120. a. What is the value of k for radioactive iodine?
Health The half-life of radioactive iodine used in medical studies is 8 hours.
a. What is the value of k for radioactive iodine?
b. A doctor wants to know when the amount of radioactive iodine in a patient’s body is 20% of the original amount. When will this occur?
Answer:
Answer: about 19 hours later
Example 6-2e

121. Multiple-Choice Test Item The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?
A 4 years B 5 years
C 20 years D 23 years
Read the Test Item
You want to know when the population has doubled or is 2 million. Use the formula
Example 6-3a

122. Exponential growth formula
Solve the Test Item
Exponential growth formula
Replace y with 2,000,000, a with 1,000,000, and r with 0.03.
Divide each side by 1,000,000.
Property of Equality for Logarithms
Power Property of Logarithms
Example 6-3b

123. Divide each side by ln 1.03.
Use a calculator.
Answer: D
Example 6-3c

124. Multiple-Choice Test Item The population of a city of 10,000 is increasing at a rate of 5% per year. If the population continues to grow at this rate, in how many years will the population have doubled?
A 10 years B 12 years
C 14 years D 18 years
Answer: C
Example 6-3d

125. You want to find t such that
Population As of 2000, Nigeria had an estimated population of 127 million people and the United States had an estimated population of 278 million people. The growth of the populations of Nigeria and the United States can be modeled by and , respectively. According to these models, when will Nigeria’s population be more than the population of the United States?
You want to find t such that
Replace N(t) with and U(t) with
Example 6-4a

126. Property of Inequality for Logarithms
Product Property of Logarithms
Inverse Property of Exponents and Logarithms
Subtract ln 278 and t from each side.
Divide each side by –0.017.
Use a calculator.
Answer: After 46 years or in 2046, Nigeria’s population will be greater than the population of the U.S.
Example 6-4b

127. Answer: after 109 years or in the year 2109
Population As of 2000, Saudi Arabia had an estimated population of 20.7 million people and the United States had an estimated population of 278 million people. The growth of the populations of Saudi Arabia and the United States can be modeled by and , respectively. According to these models, when will Saudi Arabia’s population be more than the population of the United States?
Answer: after 109 years or in the year 2109
Example 6-4c

128. End of Lesson 6

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