This document discusses evaluating and graphing logarithmic functions. It defines logarithms and explains how to evaluate logarithmic expressions without a calculator by rewriting them as exponential expressions. Examples are provided such as log 3 81 = 4 and log 2 (1/32) = -5. The definitions of common (base 10), natural (base e), and general logarithmic functions are given. Graphs of logarithmic functions are asymptotic to the horizontal axis and shift right or left depending on the base.

1. 7.4 Logarithms p. 499 What you should learn: Goal 1 Goal 2 Evaluate logarithms Graph logarithmic functions 7.4 Evaluate Logarithms and Graph Logarithmic Functions A3.2.2

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

19. Assignment