1. 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. Slide 3- 4 Quick Review

5. Slide 3- 5 Quick Review Solutions

6. 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. Slide 3- 7 Exponential Functions

8. Slide 3- 8 Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

9. Slide 3- 9 Use the rules for exponents to solve for x 4 x = 128 (2) 2x = 2 7 2x = 7 x = 7/2 2 x = 1/32 2 x = 2 -5 x = -5 Exponential Functions

10. Slide 3- 10 (x 3 y 2/3 ) 1/2 x 3/2 y 1/3 27 x = 9 -x+1 (3 3 ) x = (3 2 ) -x+1 3 3x = 3 -2x+2 3x = -2x+ 2 5x = 2 x = 2/5 Exponential Functions

11. Slide 3- 11 Example Finding an Exponential Function from its Table of Values

12. Slide 3- 12 Example Finding an Exponential Function from its Table of Values

13. Slide 3- 13 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = 2 x If b > 1, then the graph of b x will: Rise from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

14. Slide 3- 14 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = ( 1 / 2 ) x If 0 < b < 1, then the graph of b x will: Fall from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

15. Slide 3- 15 Exponential Growth and Decay

16. Slide 3- 16 Example Transforming Exponential Functions

17. Slide 3- 17 Example Transforming Exponential Functions

18. Slide 3- 18 Example Transforming Exponential Functions

19. Slide 3- 19 Example Transforming Exponential Functions

20. Slide 3- 20 The Natural Base e

21. Slide 3- 21 Exponential Functions and the Base e

22. Slide 3- 22 Exponential Functions and the Base e

23. Slide 3- 23 Example Transforming Exponential Functions

24. Slide 3- 24 Example Transforming Exponential Functions

25. Slide 3- 25 Logistic Growth Functions

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

27. Slide 3- 27 Quick Review

28. Slide 3- 28 Quick Review Solutions

29. Slide 3- 29 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.

30. Slide 3- 30 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.

31. Slide 3- 31 Exponential Population Model

32. Slide 3- 32 Example Finding Growth and Decay Rates

33. Slide 3- 33 Example Finding Growth and Decay Rates

34. Slide 3- 34 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

35. Slide 3- 35 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

36. Slide 3- 36 Example Modeling Bacteria Growth

37. Slide 3- 37 Example Modeling Bacteria Growth

38. Slide 3- 38 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

39. Slide 3- 39 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

40. Slide 3- 40 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.

41. Slide 3- 41 Example Modeling a Rumor

42. Slide 3- 42 Example Modeling a Rumor

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

44. Slide 3- 44 Quick Review

45. Slide 3- 45 Quick Review Solutions

46. Slide 3- 46 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.

47. Slide 3- 47 Changing Between Logarithmic and Exponential Form

48. Slide 3- 48 Inverses of Exponential Functions

49. Slide 3- 49 5.4 Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

50. Slide 3- 50 Basic Properties of Logarithms

51. Slide 3- 51 An Exponential Function and Its Inverse

52. Slide 3- 52 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.

53. Slide 3- 53 Basic Properties of Common Logarithms

54. Slide 3- 54 Example Solving Simple Logarithmic Equations

55. Slide 3- 55 Example Solving Simple Logarithmic Equations

56. Slide 3- 56 Basic Properties of Natural Logarithms

57. Slide 3- 57 Graphs of the Common and Natural Logarithm

58. Slide 3- 58 Example Transforming Logarithmic Graphs

59. Slide 3- 59 Example Transforming Logarithmic Graphs

60. Slide 3- 60 Decibels

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

62. Slide 3- 62 Quick Review

63. Slide 3- 63 Quick Review Solutions

64. Slide 3- 64 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.

65. Slide 3- 65 Properties of Logarithms

66. Slide 3- 66 Example Proving the Product Rule for Logarithms

67. Slide 3- 67 Example Proving the Product Rule for Logarithms

68. Slide 3- 68 Example Expanding the Logarithm of a Product

69. Slide 3- 69 Example Expanding the Logarithm of a Product

70. Slide 3- 70 Example Condensing a Logarithmic Expression

71. Slide 3- 71 Example Condensing a Logarithmic Expression

72. Slide 3- 72 Change-of-Base Formula for Logarithms

73. Slide 3- 73 Example Evaluating Logarithms by Changing the Base

74. Slide 3- 74 Example Evaluating Logarithms by Changing the Base

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

76. Slide 3- 76 Quick Review

77. Slide 3- 77 Quick Review Solutions

78. Slide 3- 78 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.

79. Slide 3- 79 One-to-One Properties

80. Slide 3- 80 Example Solving an Exponential Equation Algebraically

81. Slide 3- 81 Example Solving an Exponential Equation Algebraically

82. Slide 3- 82 Example Solving a Logarithmic Equation

83. Slide 3- 83 Example Solving a Logarithmic Equation

84. Slide 3- 84 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.

85. Slide 3- 85 Richter Scale

86. Slide 3- 86 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.

87. Slide 3- 87 Newton’s Law of Cooling

88. Slide 3- 88 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?

89. Slide 3- 89 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?

90. Slide 3- 90 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

91. Slide 3- 91 Three Types of Logarithmic Re-Expression

92. Slide 3- 92 Three Types of Logarithmic Re-Expression (cont’d)

93. Slide 3- 93 Three Types of Logarithmic Re-Expression (cont’d)

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

95. Slide 3- 95 Quick Review

96. Slide 3- 96 Quick Review Solutions

97. Slide 3- 97 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!

98. Slide 3- 98 Interest Compounded Annually

99. Slide 3- 99 Interest Compounded k Times per Year

100. Slide 3- 100 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

101. Slide 3- 101 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

102. Slide 3- 102 Compound Interest – Value of an Investment

103. Slide 3- 103 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

104. Slide 3- 104 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

105. Slide 3- 105 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.

106. Slide 3- 106 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

107. Slide 3- 107 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

108. Slide 3- 108 Future Value of an Annuity

109. Slide 3- 109 Present Value of an Annuity

110. Slide 3- 110 Chapter Test

111. Slide 3- 111 Chapter Test

112. Slide 3- 112 Chapter Test

113. Slide 3- 113 Chapter Test Solutions

114. Slide 3- 114 Chapter Test Solutions

115. Slide 3- 115 Chapter Test Solutions