1. 8.4 Logarithms p. 486

2. Evaluating Log Expressions We know 2 2 = 4 and 2 3 = 8 But for what value of y does 2 y = 6? Because 2 2 <6<2 3 you would expect the answer to be between 2 & 3. To answer this question exactly, mathematicians defined logarithms.

3. Definition of Logarithm to base a Let a & x be positive numbers & a ≠ 1. The logarithm of x with base a is denoted by log a x and is defined: log a x = y iff a y = x This expression is read “log base a of x” The function f(x) = log a x is the logarithmic function with base a.

4. The definition tells you that the equations log a x = y and a y = x are equivilant. Rewriting forms: To evaluate log 3 9 = x ask yourself… “Self… 3 to what power is 9?” 3 2 = 9 so…… log 3 9 = 2

5. Log form Exp. form log 2 16 = 4 log 10 10 = 1 log 3 1 = 0 log 10.1 = -1 log 2 6 ≈ 2.585 2 4 = 16 10 1 = 10 3 0 = 1 10 -1 =.1 2 2.585 = 6

6. Evaluate without a calculator log 3 81 = Log 5 125 = Log 4 256 = Log 2 (1/32) = 3 x = 81 5 x = 125 4 x = 256 2 x = (1/32) 4 3 4 -5

7. Evaluating logarithms now you try some! Log 4 16 = Log 5 1 = Log 4 2 = Log 3 (-1) = (Think of the graph of y=3 x ) 2 0 ½ ( because 4 1/2 = 2) undefined

8. You should learn the following general forms!!! Log a 1 = 0 because a 0 = 1 Log a a = 1 because a 1 = a Log a a x = x because a x = a x

9. Natural logarithms log e x = ln x ln means log base e

10. Common logarithms log 10 x = log x Understood base 10 if nothing is there.

11. Common logs and natural logs with a calculator log 10 button ln button

12. g(x) = log b x is the inverse of f(x) = b x f(g(x)) = x and g(f(x)) = x Exponential and log functions are inverses and “undo” each other

13. So: g(f(x)) = log b b x = x f(g(x)) = b log b x = x 10 log2 = Log 3 9 x = 10 logx = Log 5 125 x = 2 Log 3 (3 2 ) x =Log 3 3 2x =2x x 3x

14. Finding Inverses Find the inverse of: y = log 3 x By definition of logarithm, the inverse is y=3 x OR write it in exponential form and switch the x & y! 3 y = x 3 x = y

15. Finding Inverses cont. Find the inverse of : Y = ln (x +1) X = ln (y + 1) Switch the x & y e x = y + 1 Write in exp form e x – 1 = y solve for y

16. Assignment

17. Graphs of logs y = log b (x-h)+k Has vertical asymptote x=h The domain is x>h, the range is all reals If b>1, the graph moves up to the right If 0<b<1, the graph moves down to the right

18. Graph y = log 1/3 x-1 Plot (1/3,0) & (3,-2) Vert line x=0 is asy. Connect the dots X=0

19. Graph y =log 5 (x+2) Plot easy points (-1,0) & (3,1) Label the asymptote x=-2 Connect the dots using the asymptote. X=-2

20. Assignment