1.  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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 5.3 Logarithmic Functions and Graphs

2. Logarithmic Function, Base a Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.”

3. Logarithms Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley log a 1 = 0 and log a a = 1, for any logarithmic base a. The definition of a logarithm

4. Example – Using the Definition Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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

5. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Convert each of the following to a logarithmic equation. a) 16 = 2 x b) 10 –3 = 0.001c) e t = 70

6. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Convert each of the following to an exponential equation. a) log 2 32= 5b) log a Q= 8c) x = log t M

7. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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

8. Natural Logarithms Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

9. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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

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

11. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

12. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

13. Graphs of Logarithmic Functions Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

14. Graphs of Logarithmic Functions - Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graph: y = f (x) = log 5 x.

15. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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) =

16. Application Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. 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.