8.4 Logarithms

Objectives/Assignment Evaluate logarithmic functions Assignment: 17-75 odd

Evaluating Log Expressions We know 22 = 4 and 23 = 8 But for what value of y does 2y = 6? Because 22<6<23 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 logax and is defined: logax = y iff ay = x This expression is read “log base a of x” The function f(x) = logax is the logarithmic function with base a.

The definition tells you that the equations logax = y and ay = x are equivilant. Rewriting forms: To evaluate log3 9 = x ask yourself… “Self… 3 to what power is 9?” 32 = 9 so…… log39 = 2

log216 = 4 log1010 = 1 log31 = 0 log10 .1 = -1 log2 6 ≈ 2.585 24 = 16 Log form Exp. form log216 = 4 log1010 = 1 log31 = 0 log10 .1 = -1 log2 6 ≈ 2.585 24 = 16 101 = 10 30 = 1 10-1 = .1 22.585 = 6

Evaluate without a calculator 3x = 81 5x = 125 4x = 256 2x = (1/32) log381 = Log5125 = Log4256 = Log2(1/32) = 4 3 4 -5

Evaluating logarithms Now you try some! 2 Log 4 16 = Log 5 1 = Log 4 2 = Log 3 (-1) = (Think of the graph of y=3x) ½ (because 41/2 = 2) undefined

You should learn the following general forms: Log a 1 = 0 because a0 = 1 Log a a = 1 because a1 = a Log a ax = x because ax = ax

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

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

Common logs and natural logs with a calculator log10 button ln button

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

So: g(f(x)) = logbbx = x 10log2 = Log39x = 10logx = Log5125x = 2 f(g(x)) = blogbx = x 10log2 = Log39x = 10logx = Log5125x = 2 Log3(32)x = Log332x= 2x x 3x

y = log3x Finding Inverses Find the inverse of: By definition of logarithm, the inverse is y=3x OR write it in exponential form and switch the x & y! 3y = x 3x = y

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

y = logb(x-h)+k Graphs of logs 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 = log1/3x-1 Plot (1/3,0) & (3,-2) Vert line x=0 is asy. Connect the dots X=0

Graph y =log5(x+2) Plot easy points (-1,0) & (3,1) Label the asymptote x=-2 Connect the dots using the asymptote. X=-2