1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3 Exponential, Logistic, and Logarithmic Functions

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 Exponential and Logistic Functions

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 What you’ll learn about Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Exponential Functions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Example Finding an Exponential Function from its Table of Values

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 Example Finding an Exponential Function from its Table of Values

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 10 Exponential Growth and Decay

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 11 Example Transforming Exponential Functions

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 12 Example Transforming Exponential Functions

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 13 Example Transforming Exponential Functions

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 14 Example Transforming Exponential Functions

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 15 The Natural Base e

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 16 Exponential Functions and the Base e

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 17 Exponential Functions and the Base e

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 18 Example Transforming Exponential Functions

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 19 Example Transforming Exponential Functions

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 20 Logistic Growth Functions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.2 Exponential and Logistic Modeling

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 23 Quick Review Solutions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 24 What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 25 Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 26 Exponential Population Model

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 27 Example Finding Growth and Decay Rates

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 28 Example Finding Growth and Decay Rates

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 29 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 30 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 31 Example Modeling Bacteria Growth

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 32 Example Modeling Bacteria Growth

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 33 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 34 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 35 Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 36 Example Modeling a Rumor

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 37 Example Modeling a Rumor

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.3 Logarithmic Functions and Their Graphs

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 39 Quick Review

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 40 Quick Review Solutions

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 41 What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 42 Changing Between Logarithmic and Exponential Form

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 43 Inverses of Exponential Functions

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 44 Basic Properties of Logarithms

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 45 An Exponential Function and Its Inverse

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 46 Common Logarithm – Base 10 Logarithms with base 10 are called common logarithms. The common logarithm log 10 x = log x. The common logarithm is the inverse of the exponential function y = 10 x.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 47 Basic Properties of Common Logarithms

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 48 Example Solving Simple Logarithmic Equations

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 49 Example Solving Simple Logarithmic Equations

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 50 Basic Properties of Natural Logarithms

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 51 Graphs of the Common and Natural Logarithm

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 52 Example Transforming Logarithmic Graphs

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 53 Example Transforming Logarithmic Graphs

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 54 Decibels

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.4 Properties of Logarithmic Functions

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 56 Quick Review

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 57 Quick Review Solutions

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 58 What you’ll learn about Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data … and why The applications of logarithms are based on their many special properties, so learn them well.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 59 Properties of Logarithms

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 60 Example Proving the Product Rule for Logarithms

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 61 Example Proving the Product Rule for Logarithms

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 62 Example Expanding the Logarithm of a Product

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 63 Example Expanding the Logarithm of a Product

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 64 Example Condensing a Logarithmic Expression

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 65 Example Condensing a Logarithmic Expression

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 66 Change-of-Base Formula for Logarithms

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 67 Example Evaluating Logarithms by Changing the Base

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 68 Example Evaluating Logarithms by Changing the Base

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.5 Equation Solving and Modeling

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 70 Quick Review

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 71 Quick Review Solutions

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 72 What you’ll learn about Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression … and why The Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 73 One-to-One Properties

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 74 Example Solving an Exponential Equation Algebraically

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 75 Example Solving an Exponential Equation Algebraically

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 76 Example Solving a Logarithmic Equation

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 77 Example Solving a Logarithmic Equation

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 78 Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 79 Richter Scale

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 80 pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 81 Newton’s Law of Cooling

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 82 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 83 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 84 Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + blnx Exponential regression:y = a·b x Power regression:y = a·x b

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 85 Three Types of Logarithmic Re-Expression

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 86 Three Types of Logarithmic Re-Expression (cont’d)

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 87 Three Types of Logarithmic Re-Expression (cont’d)

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.6 Mathematics of Finance

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 89 Quick Review

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 90 Quick Review Solutions

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 91 What you’ll learn about Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 92 Interest Compounded Annually

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 93 Interest Compounded k Times per Year

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 94 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 95 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 96 Compound Interest – Value of an Investment

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 97 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 98 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 99 Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 100 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 101 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 102 Future Value of an Annuity

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 103 Present Value of an Annuity

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 104 Chapter Test

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 105 Chapter Test

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 106 Chapter Test

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 107 Chapter Test Solutions

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 108 Chapter Test Solutions

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 109 Chapter Test Solutions