2. Logarithmic Functions Section 3-2

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 BIG PICTURE Logarithms are just another way to write an exponential expression Cannot take Log of ZERO or any NEGATIVE #

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Exponential Equation

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Logarithmic Equation

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Conversion between Formats

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Conversions Logarithmic Equations log 5 25 = 2 log 8 1 = 0 log 9 3 = ½ log 10 100,000 = 5 Exponential Format 5 2 = 25 8 0 = 1 9 ½ = 3 10 5 = 100,000

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 log 10 –4LOG –4 ENTERERROR no power of 10 gives a negative number Common Logarithmic Function The base 10 logarithm function f (x) = log 10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. log 10 100 log 10 5 Function ValueKeystrokesDisplay LOG 100 ENTER2 LOG 5 ENTER0.6989700 log 10 ( ) – 0.3979400LOG ( 2 5 ) ENTER

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 x y Graph f(x) = log 2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. 83 42 21 10 –1 –2 2x2x x y = log 2 x y = x y = 2 x (1, 0) x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: f(x) = log 0 x Example: Graph the common logarithm function f(x) = log 10 x. by calculator 10.6020.3010–1–2f(x) = log 10 x 10421x y x 5 –5 f(x) = log 10 x x = 0 vertical asymptote (0, 1) x-intercept

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = log a x (a  1) 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x 1. domain 2. range 4. vertical asymptote Graph of f (x) = log a x (a  1) x y y = x y = log 2 x y = a x domain range y-axis vertical asymptote x-intercept (1, 0)

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Natural Logarithmi c Function The function defined by f(x) = log e x = ln x is called the natural logarithm function. Use a calculator to evaluate: ln 3, ln –2, ln 100 ln 3 ln –2 ln 100 Function ValueKeystrokesDisplay LN 3 ENTER1.0986122 ERRORLN –2 ENTER LN 100 ENTER4.6051701 y = ln x (x  0, e 2.718281  ) y x 5 –5 y = ln x is equivalent to e y = x

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Log Functions Overview Log FunctionBase log a xa log x10 log e xe ln xe

14. 13 Homework WS 6-2 Quiz next class on graphing