8.4 Logarithms p. 486

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.

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.

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

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

Evaluate without a calculator log 3 81 = Log = Log = Log 2 (1/32) = 3 x = 81 5 x = x = x = (1/32)

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

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

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

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

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

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

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 x = 2 Log 3 (3 2 ) x =Log 3 3 2x =2x x 3x

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

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

Assignment

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

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

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

Assignment