1. Logarithmic Functions

2. Logarithm = Exponent Very simply, a logarithm is an exponent of ten that will produce the desired number. Y = Log 100 means what is the exponent of 10 which will produce 100. Y = log.001 means what is the exponent of 10 which will produce.001. Note: when no base is indicated, the base is 10.

3. Y=log b a can be read “ y is the exponent of base b to produce a.” 3= log 2 8 is read “3 is the exponent of 2 to produce 8” 2 = log 4 16 4 2 = 16

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Examples: Write Equivalent Equations y = log 2 ( ) = 2 y Write the equivalent exponential equation and solve for y. 1 = 5 y y = log 5 1 16 = 4 y y = log 4 16 16 = 2 y y = log 2 16 SolutionEquivalent Exponential Equation Logarithmic Equation 16 = 2 4  y = 4 = 2 -1  y = –1 16 = 4 2  y = 2 1 = 5 0  y = 0

5. Definition of a Logarithm A logarithm, or log, is defined in terms of an exponent. If b x =a, then log b a =x If 5 2 =25 then log 5 25=2 Log 5 25=2 is read “log base 5 of 25 is 2.” – You might say the log is the exponent we apply to 5 to make 25

6. Log of a Product The log of a product is the sum of the logs of the factors log b xy = log b x + log b y Log 2 512 = log 2 (8·64) = log 2 8 + log 2 64 = 3 + 5 = 8

7. Log of a Quotient The log of a quotient is the difference of the logs of the factors. Ex.

8. Log of a Power The log of a power is the product of the exponent and the log of the base. log b x n = nlogbx Ex: log 3 2 = 2log3

9. Use the properties of logs to simplify the following: Given:

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Definition: Logarithmic Function For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 log 10 –4 LOG –4 ENTER ERROR REMEMBER: no power of 10 gives a negative number Common Logarithmic Function The logarithm function f (x) = log 10 x is called the common logarithm function. log 10 100 log 10 5 Function Value Keystrokes Display LOG 100 ENTER 2 LOG 5 ENTER 0.6989700 log 10 ( ) –.3979400 LOG ( 2 / 5 ) ENTER

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Properties of Logarithms 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 3. If log a x = log a y, then x = y. one-to-one property

13. INVERSE PROPERTIES: The logarithm with base a of a raised to a power equals that power: a raised to the logarithm with base a of a number equals that number Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Graph f(x) = log 2 x One way to 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 y x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

15. Graphing Log Functions

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Graph f(x) = log 2 x One way to 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 y x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Example: f(x) = log 0 x Graph of 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

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 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 asympto te x-intercept (1, 0)

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Using the graphing calculator to graph the log functions For other than base 10, use the following formula: To graph the function

20. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Graph the following functions:

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Natural Logarith mic 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 Value KeystrokesDisplay LN 3 ENTER1.0986122 ERRORLN –2 ENTER LN 100 ENTER4.6051701 y = ln x (x  0, e.718281  ) y x 5 –5 y = ln x is equivalent to e y = x

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 1. 2. ln e = 1 since e 1 = e. 3. ln e x = x and e ln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property property 2 property 1

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