8.4 Logarithms

By: L. Keali’i Alicea

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

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)

4

34

-5

Evaluating logarithms now you try some!

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

20

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

Common logarithms

•log 10 x = log x

•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• f(g(x)) = blog

bx = x

•10log2 = •Log39x =•10logx =•Log5125x =

2Log3(32)x =Log332x=2x

x3x

Finding Inverses

• 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 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

Assignment

Graphs of logs

•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

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

8.4 A (all)

8.4 B (1-27 odd, 28-29)