1. 7.4A Logarithms
Algebra II

2. 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.

3. 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.

4. 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

5. 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
= 6

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

7. 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

8. 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

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

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

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

12. 7.4B Finding Inverses & Graphs
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

13. 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

14. y = log3x Ex. 1)Finding Inverses 3x = y Find the inverse of:
( write it in exponential form and switch the x & y! )
3y = x
3x = y

15. Ex. 2) Finding Inverses cont.
Find the inverse of :
Y = ln (x +1)
X = ln (y + 1) Switch the x & y
ex = y Write in exp form
ex – 1 = y solve for y

16. 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

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

18. Ex. 4)

19. Ex. 5) Graph y = log1/3x-1 (without parentheses)
Plot (1/3,0) & (3,-2)
Vert line x=0 is asy.
Connect the dots
X=0

20. Assignment