1. 5.2 L OGARITHMIC F UNCTIONS & T HEIR G RAPHS Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real- life problems.

2. Is this function one to one? Horizontal Line test? Does it have an inverse? f(x) = 3 x

3. L OGARITHMIC FUNCTION WITH BASE “ A ” Definition For x > 0, a > 0, and a  1, y = log a x if and only if x = a y The function given by f(x) = log a x read as “log base a of x” is called the logarithmic function with base a.

4. W RITING THE LOGARITHMIC EQUATION IN EXPONENTIAL FORM log 3 81 = 4 log 16 8 = 3 / 4 8 2 = 64 4 -3 = 1 / 64 3 4 = 8116 3/4 = 8 log 8 64 = 2 log 4 ( 1 / 64 ) = -3 W RITING AN EXPONENTIAL EQUATION IN LOGARITHMIC FORM

5. E VALUATING L OGS y= log 2 32 y= log 4 2 y= log 3 1 y= log 10 1 / 100 Step 1: Rewrite the log problem as an exponential. Step 2: Rewrite both sides of the = with the same base. 2 y = 32 2 y = 2 5 Therefore, y = 5 y = 5 4 y = 2 (2 2 ) y = 2 1 2 2y = 2 1 y = 1 / 2 3 y = 1 y = 0 10 y = 1 / 100 y = -2 10 y = 10 -2 y = -2

6. E VALUATING L OGS ON A C ALCULATOR f(x) = log x when x = 10 f(x) = 1 when x = 1/3 f(x) = -.4771 when x = 2.5 f(x) =.3979 when x = -2 f(x) = ERROR!!! Why??? You can only use a calculator when the base is 10

7. P ROPERTIES OF L OGARITHMS log a 1 = 0 because a 0 = 1 log a a = 1 because a 1 = a log a a x = x and a log a x = x log a x = log a y, then x = y

8. S IMPLIFY USING THE PROPERTIES OF LOGS log 4 1 log  7  7 6 log 6 20 Rewrite as an exponent 4 y = 1 So y = 0 Rewrite as an exponent  7 y =  7 So y = 1

9. U SE THE 1-1 PROPERTY TO SOLVE log 3 x = log 3 12 log 3 (2x + 1) = log 3 x log 4 (x 2 - 6) = log 4 10 x = 12 2x + 1 = x x = -1 x 2 - 6 = 10 x 2 = 16 x =  4

10. F ( X ) = 3 X Graphs of Logarithmic Functions So, the inverse would be g(x) = log 3 x Make a T chart Domain— Range? Asymptotes?

11. Graphs of Logarithmic Functions g(x) = log 4 (x – 3) Make a T chart Domain— Range? Asymptotes?

12. Graphs of Logarithmic Functions g(x) = log 5 (x – 1) + 4 Make a T chart Domain— Range? Asymptotes?

13. N ATURAL L OGARITHMIC F UNCTIONS The function defined by f(x) = log e x = ln x, x > 0 is called the natural logarithmic function.

14. E VALUATE f(x) = ln x when x = 2 f(x) =.6931 when x = -1 f(x) = Error!!! Why???

15. P ROPERTIES OF N ATURAL L OGARITHMS ln 1 = 0 because e 0 = 1 ln e = 1 because e 1 = e ln e x = x and e lnx = x (Think…they are inverses of each other.) If ln x = ln y, then x = y

16. U SE PROPERTIES OF N ATURAL L OGS TO SIMPLIFY EACH EXPRESSION ln (1/e) = ln e -1 = -1 e ln 5 = 5 2 ln e = 2

17. Graphs of Natural Logs g(x) = ln(x + 2) Make a T chart Domain— Range? Asymptotes? 2 Undefined 3 4

18. Graphs of Natural Logs g(x) = ln(2 - x) Make a T chart Domain— Range? Asymptotes? 2 Undefined 1 0