p. 486p. 486

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.

Evaluating Log Expressions

Let a & x be positive numbers & a ≠ 1. The logarithm of x with base a is denoted by

logax and is defined:

logax = y if ay = x This expression is read “log base a of x” The function f(x) = logax is the logarithmic

function with base a.

Definition of Logarithm to 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

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

log381 = Log5125 = Log4256 = Log2(1/32) =

3x = 81 5x = 125 4x = 256 2x = (1/32)

434

-5

Log 4 16 = Log 5 1 = Log 4 2 = Log 3 (-1) = (Think of the graph of y=3x)

Evaluating logarithms now you try some!

20½ (because 41/2 = 2)

undefined

Log a 1 = 0 because a0 = 1

Log a a = 1 because a1 = a

Log a ax = x because ax = ax

You should learn the following general forms!!!

log e x = ln xln means log base e

Natural logarithms

log 10 x = log x

Understood base 10 if nothing is there.

Common logarithms

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

f(g(x)) = blogbx = x

10log2 = Log39x = 10logx = Log5125x =

2Log3(32)x =Log332x=2x

x3x

Find the inverse of: y = log3x By definition of logarithm, the

inverse is y=3x OR write it in exponential form and

switch the x & y! 3y = x 3x = y

Finding Inverses

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

Finding Inverses cont.

Assignment

y = logb(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

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

Assignment