1. The Logarithmic Functions and Their Graphs Section 3.2

2. Objectives: Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions with base a. Recognize, evaluate, and graph logartithmic functions with base e.

3. 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. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

4. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic form Convert to log form: Convert to exponential form: When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.

5. LOGS = EXPONENTS With this in mind, we can answer questions about the log: This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?) What exponent do you put on the base of 3 to get 1/9? (hint: think negative) What exponent do you put on the base of 4 to get 1? When working with logs, re-write any radicals as rational exponents. What exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)

6. Examples: Write Equivalent Equations y = log 2 ( ) = 2 y Examples: 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

7. Your Turn: Write each equation in exponential form –log 3 81 = 4 –3 4 =81 –log 7 1/49 = -2 –7 -2 =1/49 Write each equation in logarithmic form –10 3 = 1000 –Log 10 1000=3 –4 -2 = 1/16 –Log 4 1/16=-2

8. Your Turn: Find y in each equation. log 2 8 = y –y=3 log 5 1 = y 25 –y=-2

9. Your Turn: Find a in each equation. log a 36 = 2 –a=6 log 8 a = -1/3 –a=1/2

10. Properties of Logarithms Examples: Solve for x: log 6 6 = x log 6 6 = 1 property 2  x = 1 Simplify: log 3 3 5 log 3 3 5 = 5 property 3 Simplify: 7 log 7 9 7 log 7 9 = 9 property 3 Properties of Logarithms 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 4. If log a x = log a y, then x = y. one-to-one property 3. log a a x = x and a log a x = x inverse property

11. Your Turn: Solve 1. –0–0 2. –1–1 3. –30

12. Base 10 logarithms Called common logarithms When base a is not indicated, it is understood that a = 10 log 1/100 =log 10 = log 1/10 =log 100 = log 1 =log 1000 = The LOG key on your calculator. -2 0 1 2 3

13. In the last section we learned about the graphs of exponentials. Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials. Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa. Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.

14. Characteristics about the Graph of an Exponential Function a > 1 1. Domain is all real numbers 2. Range is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  -  Characteristics about the Graph of a Log Function a > 1 1. Range is all real numbers 2. Domain is positive real numbers 3. There are no y intercepts 4. The x intercept is always (1,0) (x’s and y’s trade places) 5. The graph is always increasing 6. The y-axis (where x = 0) is a vertical asymptote

15. Exponential Graph Logarithmic Graph Graphs of inverse functions are reflected about the line y = x base a>1

16. Graph f(x) = log 2 x x y 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

17. 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)

18. Graphs of Logarithmic Functions Typical shape for graphs where a > 1 (includes base e and base 10 graphs). Typical shape for graphs where 0 < a < 1.

19. The Logarithmic Function: f (x) = log a x, a > 1

20. The Logarithmic Function: f (x) = log a x, 0 < a < 1

21. Determining Domains of Logarithmic Functions ExampleFind the domain of each function. Solution (a)Argument of the logarithm must be positive. x – 1 > 0, or x > 1. The domain is (1,  ). (b)Use the sign graph to solve x 2 – 4 > 0. The domain is (– ,–2) (2,  ).

22. Your Turn: Find the domain. (a) Solution a)Domain: (-3,  ) or x >-3 b)Domain: (-3, 3) or -3< x <3

23. Any log to the base e is known as a natural logarithm. In French this is a logarithme naturel Which is where ln comes from. When you see ln (instead of log) –then it’s a natural log y = ln x is the inverse of y = e x The LN key on your calculator. Natural Logarithms

29. Changing the Base a of Logarithmic Functions

30. Transformations Logarithmic Functions

31. Logarithmic Functions

34. Transformation of functions apply to log functions just like they apply to all other functions so let’s try a couple. up 2 left 1 Reflect about x axis

35. Assignment Sec. 3.2, pg. 203 – 206: #1 – 41 odd, 47 – 79 odd