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

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

8.   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 You should learn the following general forms!!!

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

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

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.   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 Finding Inverses

15.   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 Finding Inverses cont.

16.  Assignment

17.   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 Graphs of logs

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