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

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 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.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Properties of Logarithms

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 Example Proving the Product Rule for Logarithms

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Example Proving the Product Rule for Logarithms

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Example Expanding the Logarithm of a Product

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 Example Expanding the Logarithm of a Product

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 10 Example Condensing a Logarithmic Expression

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 11 Example Condensing a Logarithmic Expression

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 12 Change-of-Base Formula for Logarithms

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 13 Example Evaluating Logarithms by Changing the Base

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 14 Example Evaluating Logarithms by Changing the Base

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

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 16 Quick Review

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

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

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 19 One-to-One Properties

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 20 Example Solving an Exponential Equation Algebraically

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 21 Example Solving an Exponential Equation Algebraically

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 22 Example Solving a Logarithmic Equation

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 23 Example Solving a Logarithmic Equation

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 24 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.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 25 Richter Scale

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

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 27 Newton’s Law of Cooling

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 28 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?

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 29 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?

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 30 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

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 31 Three Types of Logarithmic Re-Expression

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 32 Three Types of Logarithmic Re-Expression (cont’d)

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 33 Three Types of Logarithmic Re-Expression (cont’d)

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

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

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 36 Quick Review Solutions

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 37 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!

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 38 Interest Compounded Annually

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 39 Interest Compounded k Times per Year

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 40 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 41 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 42 Compound Interest – Value of an Investment

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 43 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 44 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

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

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 46 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 47 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 48 Future Value of an Annuity

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 49 Present Value of an Annuity

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 50 Chapter Test

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 51 Chapter Test

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

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

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

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 55 Chapter Test Solutions