1. 4.4 Evaluate Logarithms & Graph Logarithmic Functions
What is a logarithm? How do you read it?
What relationship exists between logs and exponents? What is the definition?
How do you rewrite log equations?
What are two special log values?
What is a common log? A natural log?
What logs can you evaluate using a calculator?

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 b
Let b & x be positive numbers & b ≠ 1.
The logarithm of x with base a is denoted by logbx and is defined:
logbx = y if by = x
This expression is read “log base b of x”
The function f(x) = logbx is the logarithmic function with base b.

4. The definition tells you that the equations logbx = y and by = x are equivalent.
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. 3x = 81 5x = 125 4x = 256 2x = (1/32) log381 = Log5125 = Log4256 =
Evaluate
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)
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8. You should learn the following general forms!!!
Log b 1 = 0 because b0 = 1
Log b b = 1 because b1 = b
Log b bx = x because bx = bx

9. Natural logarithms
log e x = ln x
ln means log base e 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
**Only common log and natural log bases are on a calculator.

12. Keystrokes Expression Keystrokes Display Check a. log 8 8 0.903089987 8 b.
ln 0.3 .3 –
0.3
e –1.204

13. Tornadoes
The wind speed s (in miles per hour) near the center of a tornado can be modeled by:
where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center.
93 log d + 65 s =

14. Solution 93 log d + 65 s = Write function. = 93 log 220 + 65
Substitute 220 for d.
93(2.342) + 65
Use a calculator. =
Simplify.
The wind speed near the tornado’s center was about 283 miles per hour.
ANSWER

15. What is a logarithm? How do you read it?
A logarithm is another way of expressing an exponent. It is read log base b of y.
What relationship exists between logs and exponents? What is the definition?
logax = y if ay = x
How do you rewrite logs?
The base with the exponent on the other side of the = .
What are two special log values?
Logb1=0 and logbb=1
What is a common log? A natural log?
Common log is base 10. Natural log is base e.
What logs can you evaluate using a calculator?
Base 10

16. 4.4 Assignment
Page 255, 3-6, 8-16, even

17. 4.4 Day 2 How do you use inverse properties with logarithms?
How do you graph logs?

18. 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.

19. 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

20. Use Inverse Properties
Simplify the expression. a.
10log4 b. 5
log
25x
SOLUTION a.
10log4 = 4 b
log x
= x b. 5
log
25x = ( 52 ) x 5
log
Express 25 as a power with base 5. = 5
log
52x
Power of a power property 2x = b
log bx
= x

21. Find Inverse Properties
Find the inverse of the function. a.
y = 6 x b.
y = ln (x + 3)
SOLUTION a. 6
log
From the definition of logarithm, the inverse of
y = 6 x is y= x. b.
y = ln (x + 3)
Write original function.
x = ln (y + 3)
Switch x and y. = ex
(y + 3)
Write in exponential form. =
ex – 3 y
Solve for y.
ANSWER
The inverse of y = ln (x + 3) is y =
ex – 3.

22. Use Inverse Properties
Simplify the expression.
10. 8
log x
SOLUTION b
log b
= x
Exponent form 8
log x = x
Log form
11. 7
log
7–3x a
log ax
= x
SOLUTION
Log form 7
log
7–3x =
–3x
Exponent form

23. Use Inverse Properties
Find the inverse of
14.
y = 4 x
SOLUTION
From the definition of logarithm, the inverse of 4
log
y = x. is
y = ln (x – 5).
Find the inverse of
15.
SOLUTION
y = ln (x – 5)
Write original function.
x = ln (y – 5)
Switch x and y. = ex
(y – 5)
Write in exponential form. =
ex + 5 y
Solve for y.
ANSWER
The inverse of y = ln (x – 5) is y =
e x + 5.

24. 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

25. 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 Write in exp form
ex – 1 = y solve for y

26. 4.4 Graphing Logs
p. 254

27. 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

28. Graphing a log function
Graph the function. a. y = 3
log x
SOLUTION
Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote.
From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

29. Graph y = log1/3x-1 Plot (1/3,0) & (3,-2) Vert line x=0 is asy.
Connect the dots
X=0

30. 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

31. How do you use inverse properties with logarithms?
Exponential and log functions are inverses and “undo” each other.
How do you graph logs?
Pick 1, the base number, and a power of the base for x.

32. 4.4 Assignment Day 2
Page 256, even, odd