Logarithmic Functions: Properties & Applications

1. Homework, Pag3 308 Evaluate the logarithmic expression without using a calculator. 1.

2. Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 5.

3. Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 9.

4. Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 13.

5. Homework, Page 308 Evaluate the logarithmic expression without using a calculator. 17.

6. Homework, Page 308 Evaluate the expression without using a calculator. 21.

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. 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. Homework, Page 308 Solve the equation by changing it to exponential form. 33.

10. Homework, Page 308 Match the function with its graph. 37. d.

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. 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. 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. 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. 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. 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. 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.

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. 3.4 Properties of Logarithmic Functions

20. Quick Review

21. Quick Review Solutions

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. Leading Questions Is logb Rc = c logbR a correct statement? Does log xy = log x – log y ? Does logb R = ln R / ln b ?

24. Properties of Logarithms

25. Example Proving the Product Rule for Logarithms

26. Example Expanding the Logarithm of a Product

27. Example Expanding the Logarithm of a Quotient

28. Example Condensing a Logarithmic Expression

29. Change-of-Base Formula for Logarithms

30. Example Evaluating Logarithms by Changing the Base

31. Example Graphing Logarithmic Functions

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. 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.

34. Example Re-Expressing Kepler’s Third Law

35. Homework • Review Section 3.4 • Page 317, Exercises: 1 – 65 (EOO) • Quiz next time

36. 3.5 Equation Solving and Modeling

37. Quick Review

38. Quick Review Solutions

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. One-to-One Properties

41. Example Solving an Exponential Equation Algebraically

42. Example Solving an Exponential Equation Graphically

43. Example Solving a Logarithmic Equation Algebraically

44. Example Solving a Logarithmic Equation Graphically

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. Richter Scale

47. Example Comparing Magnitudes of Earthquakes Measured on the Richter Scale

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. 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. Newton’s Law of Cooling