1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1 Homework, Pag3 308 Evaluate the logarithmic expression without using a calculator. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 2 Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 3 Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 4 Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 Homework, Page 308 Evaluate the expression without using a calculator. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Homework, Page 308 Use a calculator to evaluate the logarithmic expression, if it is defined, and check your results by evaluating the corresponding exponential expression. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Homework, Page 308 Use a calculator to evaluate the logarithmic expression, if it is defined, and check your results by evaluating the corresponding exponential expression. 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 Homework, Page 308 Solve the equation by changing it to exponential form. 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 10 Homework, Page 308 Match the function with its graph. 37. d.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 11 Homework, Page 308 Describe how to transform the graph of y = ln x into the graph of the given function. Sketch the graph by hand and support your sketch with a grapher. 41.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 12 Homework, Page 308 Describe how to transform the graph of y = ln x into the graph of the given function. Sketch the graph by hand and support your sketch with a grapher. 45.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 13 Homework, Page 308 Describe how to transform the graph of y = log x into the graph of the given function. Sketch the graph by hand and support your sketch with a grapher. 49.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 14 Homework, Page 308 Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, boundedness, extrema, symmetry, asymptotes, and end behavior 53.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 15 Homework, Page 308 Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, boundedness, extrema, symmetry, asymptotes, and end behavior 57.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 16 Homework, Page 308 59. Use the table to compute the sound intensity in decibels for (a) a soft whisper, (b) city traffic, and (c) a jet at take-off.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 17 Homework, Page 308 61. Using the data in the table, compute a logarithmic regression model, and use it to predict when the population of San Antonio will be 1,500,000. The model predicts the population will reach 1,500,000 in 2032. YearPop.YearPop. 1970654,1531990935,933 1980785,94020001,151,305

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 18 Homework, Page 308 65. What is the approximate value of the common log of 2? a.0.10523 b.0.20000 c.0.30103 d.0.69315 e.3.32193

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

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

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

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

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leading Questions Is log b R c = c log b R a correct statement? Does log xy = log x – log y ? Does log b R = ln R / ln b ? Slide 3- 23

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

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 25 Example Proving the Product Rule for Logarithms

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 Expanding the Logarithm of a Quotient

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 Graphing Logarithmic Functions

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 32 Re-Expression of Data If we apply a function to one or both of the variables in a data set, we transform it into a more useful form, e.g., in an earlier section we let the numbers 0 – 100 represent the years 1900 – 2000. Such a transformation is called a re-expression.

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 33 Example Re-Expressing Kepler’s Third Law Re-express the (a, T) data points in Table 3.20 as (ln a, ln T) pairs. Find a linear regression model for the re-expressed pairs. Rewrite the linear regression in terms of a and T, without logarithms or fractional exponents. PlanetAvg Dist (AU)Period (years) Mercury0.38700.2410 Venus0.72330.6161 Earth1.0000 Mars1.5231.981 Jupiter5.20311.86 Saturn9.53929.46

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 34 Example Re-Expressing Kepler’s Third Law

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 35 Homework Review Section 3.4 Page 317, Exercises: 1 – 65 (EOO) Quiz next time

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

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

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 38 Quick Review Solutions

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

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 40 One-to-One Properties

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 41 Example Solving an Exponential Equation Algebraically

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 42 Example Solving an Exponential Equation Graphically

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 43 Example Solving a Logarithmic Equation Algebraically

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 44 Example Solving a Logarithmic Equation Graphically

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

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

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 47 Example Comparing Magnitudes of Earthquakes Measured on the Richter Scale

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 48 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.

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 49 Example Using pH Measurements to Compare Hydrogen Ion concentrations Compare the hydrogen ion concentrations of vinegar, with a pH of 2.4 and salt water with a pH of 7.

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 50 Newton’s Law of Cooling

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

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 52 Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + b·ln x Exponential regression:y = a·b x Power regression:y = a·x b

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 53 Three Types of Logarithmic Re-Expression

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 54 Three Types of Logarithmic Re-Expression (cont’d)

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 55 Three Types of Logarithmic Re-Expression (cont’d)