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8.4 Logarithms

- p. 486

Evaluating Log Expressions

- We know 22 4 and 23 8
- But for what value of y does 2y 6?
- Because 22lt6lt23 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

- 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 y3x)

0

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)) blogbx x
- 10log2
- Log39x
- 10logx
- Log5125x

2

Log3(32)x

Log332x

2x

x

3x

Finding Inverses

- Find the inverse of
- y log3x
- By definition of logarithm, the inverse is y3x
- 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 xh
- The domain is xgth, the range is all reals
- If bgt1, the graph moves up to the right
- If 0ltblt1, the graph moves down to the right

Graph y log1/3x-1

- Plot (1/3,0) (3,-2)
- Vert line x0 is asy.
- Connect the dots

X0

Graph y log5(x2)

- Plot easy points (-1,0) (3,1)
- Label the asymptote x-2
- Connect the dots using the asymptote.

X-2

Assignment

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