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

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 2 Quick Review

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

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

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Changing Between Logarithmic and Exponential Form

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 Inverses of Exponential Functions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Basic Properties of Logarithms

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 An Exponential Function and Its Inverse

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 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.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 10 Basic Properties of Common Logarithms

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 11 Example Solving Simple Logarithmic Equations

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 12 Example Solving Simple Logarithmic Equations

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 13 Basic Properties of Natural Logarithms

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 14 Graphs of the Common and Natural Logarithm

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 15 Example Transforming Logarithmic Graphs

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 16 Example Transforming Logarithmic Graphs

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 17 Decibels

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

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 19 Quick Review

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 20 Quick Review Solutions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 21 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.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 22 Properties of Logarithms

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 23 Example Proving the Product Rule for Logarithms

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 24 Example Proving the Product Rule for Logarithms

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 25 Example Expanding the Logarithm of a Product

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 26 Example Expanding the Logarithm of a Product

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 27 Example Condensing a Logarithmic Expression

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 28 Example Condensing a Logarithmic Expression

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 29 Change-of-Base Formula for Logarithms

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 30 Example Evaluating Logarithms by Changing the Base

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 31 Example Evaluating Logarithms by Changing the Base

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

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 33 Quick Review

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 34 Quick Review Solutions

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 35 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.

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 36 One-to-One Properties

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 37 Example Solving an Exponential Equation Algebraically

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 38 Example Solving an Exponential Equation Algebraically

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 39 Example Solving a Logarithmic Equation

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 40 Example Solving a Logarithmic Equation

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 41 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.

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 42 Richter Scale

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

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 44 Newton’s Law of Cooling

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 45 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?

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 46 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?

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 47 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

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 48 Three Types of Logarithmic Re-Expression

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 49 Three Types of Logarithmic Re-Expression (cont’d)

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 50 Three Types of Logarithmic Re-Expression (cont’d)

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

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 52 Quick Review

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 53 Quick Review Solutions

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

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 55 Interest Compounded Annually

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 56 Interest Compounded k Times per Year

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 57 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 58 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 59 Compound Interest – Value of an Investment

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 60 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 61 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 62 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.

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 63 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 64 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 65 Future Value of an Annuity

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 66 Present Value of an Annuity

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 67 Chapter Test

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 68 Chapter Test

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 69 Chapter Test

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 70 Chapter Test Solutions

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

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 72 Chapter Test Solutions