2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest

3. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 5.3 Logarithmic Functions and Graphs  Find common logarithms and natural logarithms with and without a calculator.  Convert between exponential and logarithmic equations.  Change logarithmic bases.  Graph logarithmic functions.  Solve applied problems involving logarithmic functions.

4. Slide 5.3 - 4 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Logarithmic Functions These functions are inverses of exponential functions. We can draw the graph of the inverse of an exponential function by interchanging x and y. To Graph: x = 2 y. * Note that the curve does not touch or cross the y-axis. 1.Choose values for y. 2.Compute values for x. 3.Plot the points and connect them with a smooth curve.

5. Slide 5.3 - 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Graph: x = 2 y.

6. Slide 5.3 - 6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) This curve looks like the graph of y = 2 x reflected across the line y = x, as we would expect for an inverse. The inverse of y = 2 x is x = 2 y.

7. Slide 5.3 - 7 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Logarithmic Function, Base a We define y = log a x as that number y such that x = a y, where x > 0 and a is a positive constant other than 1. We read log a x as “the logarithm, base a, of x.”

8. Slide 5.3 - 8 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Finding Certain Logarithms - Example Find each of the following logarithms. a) log 10 10,000b) log 10 0.01c) log 2 8 d) log 9 3e) log 6 1f) log 8 8 Solution: a)The exponent to which we raise 10 to obtain 10,000 is 4; thus log 10 10,000 = 4. The exponent to which we raise 10 to get 0.01 is –2, so log 10 0.01 = –2. b)We have

9. Slide 5.3 - 9 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) c) log 2 8: The exponent to which we raise 2 to get 8 is 3, so log 2 8 = 3. d) log 9 3: The exponent to which we raise 9 to get 3 is 1/2; thus log 9 3 = 1/2. e) log 6 1: 1 = 6 0. The exponent to which we raise 6 to get 1 is 0, so log 6 1 = 0. f) log 8 8: 8 = 8 1. The exponent to which we raise 8 to get 8 is 4, so log 8 8 = 1.

10. Slide 5.3 - 10 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Logarithms log a 1 = 0 and log a a = 1, for any logarithmic base a. A logarithm is an exponent!

11. Slide 5.3 - 11 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Convert each of the following to a logarithmic equation. a) 16 = 2 x b) 10 –3 = 0.001c) e t = 70 b) 10 –3 = 0.001  log 10 0.001 = –3 c) e t = 70  log e 70 = t The exponent is the logarithm. a) 16 = 2 x log 2 16 = x The base remains the same. Solution:

12. Slide 5.3 - 12 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Convert each of the following to an exponential equation. a) log 2 32= 5b) log a Q= 8c) x = log t M b) log a Q = 8  a 8 = Q c) x = log t M  t x = M The logarithm is the exponent. a) log 2 32 = 5 2 5 = 32 The base remains the same. Solution:

13. Slide 5.3 - 13 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 645,778b) log 0.0000239c) log (  3) b) log 0.0000239 –4.6216 c) log (–3) Does not exist. Solution: Function ValueReadoutRounded a) log 645,778 5.8101

14. Slide 5.3 - 14 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Natural Logarithms Logarithms, base e, are called natural logarithms. The abbreviation “ln” is generally used for natural logarithms. Thus, ln xmeans log e x. ln 1 = 0 and ln e = 1, for the logarithmic base e.

15. Slide 5.3 - 15 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 645,778b) ln 0.0000239c) log (  5) d) ln ee) ln 1 Solution: Function ValueReadoutRounded a) ln 645,778 13.3782 b) ln 0.0000239 –10.6416

16. Slide 5.3 - 16 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: Function ValueReadoutRounded c) ln 1 0b) ln e 1 c) ln (–5) Does not exist.

17. Slide 5.3 - 17 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Changing Logarithmic Bases The Change-of-Base Formula For any logarithmic bases a and b, and any positive number M,

18. Slide 5.3 - 18 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Solution: First, we let a = 10, b = 5, and M = 8. Then we substitute into the change-of-base formula: Find log 5 8 using common logarithms.

19. Slide 5.3 - 19 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Solution: Substituting e for a, 6 for b and 8 for M, we have We can also use base e for a conversion. Find log 5 8 using natural logarithms.

20. Slide 5.3 - 20 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graphs of Logarithmic Functions - Example Graph: y = f (x) = log 5 x. Solution: y = log 5 x is equivalent to x = 5 y. Select y and compute x.

21. Slide 5.3 - 21 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Graph each of the following. Describe how each graph can be obtained from the graph of y = ln x. Give the domain and the vertical asymptote of each function. a) f (x) = ln (x + 3) b) f (x) = c) f (x) = |ln (x – 1)|

22. Slide 5.3 - 22 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) a) f (x) = ln (x + 3) The graph is a shift 3 units left. The domain is the set of all real numbers greater than –3, (–3, ∞). The line x = –3 is the vertical asymptote.

23. Slide 5.3 - 23 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) b) f (x) = The graph is a vertical shrinking of y = ln x, followed by a reflection across the x-axis and a translation up 3 units. The domain is the set of all positive real numbers, (0, ∞). The y-axis is the vertical asymptote.

24. Slide 5.3 - 24 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) c) f (x) = |ln (x – 1)| The graph is a translation of y = ln x, right 1 unit. The effect of the absolute is to reflect the negative output across the x-axis. The domain is the set of all positive real numbers greater than 1, (1, ∞). The line x =1 is the vertical asymptote.

25. Slide 5.3 - 25 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Application In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05.

26. Slide 5.3 - 26 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example a.The population of Savannah, Georgia, is 132,410. Find the average walking speed of people living in Savannah. b.The population of Philadelphia, Pennsylvania, is 1,540,351. Find the average walking speed of people living in Philadelphia.

27. Slide 5.3 - 27 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: a. Since P is in thousands and 132,410 = 132.410 thousand, we substitute 132.410 for P: b. Substitute 1540.351 for P: w(1540.351) = 0.37 ln 1540.351 + 0.05  2.8 ft/sec. The average walking speed of people living in Philadelphia is about 2.8 ft/sec. w(132.410) = 0.37 ln 132.410 + 0.05  1.9 ft/sec. The average walking speed of people living in Savannah is about 1.9 ft/sec.