Download TPT 8-3-Properties-Of

8.3 Properties of logarithms
©2006 by R. Villar
All Rights Reserved
Warm-up
Find the inverse of each:
1. g(x) = 5x
2. f(x) = 2 + log4x
x = 5y
y = log5 x
Simplify each:
3. x 2 • x 5
x7
x = 2 + log4 y
x – 2 = log4 y
y = 4x – 2
4.
x8
x2
x6
5.
(x 5 )3
x 15
59. g(x) = 6x
67.
60. g(x) = log8x
68.
61. g(x) = log1/3 x
69.
62. g(x) = (1/2)x
70.
Properties of Logarithms
Consider the following two problems:
Simplify
log3 (9 • 27)
2
3
= log3 (3 • 3 )
2+3
= log3 (3 )
=
2+3
Simplify
log3 9 + log3 27
2
3
= log3 3 + log3 3
= 2 + 3
These examples suggest the following property:
Product Property of Logarithms:
For all positive numbers m, n and b where b ≠ 1,
logb mn = logb m + logb n
We will use the Product Property of Logarithms to
solve problems...
Example
Given log2 5 = 2.322, find log2 40
3
log2 20 = log2 (2 • 5)
3
= log2 2 + log2 5
= 3 + 2.322
=
5.322
Consider the following:
81 
a. log 3  
27 
4
= log3 3
3
3
4–3
= log3 3
=
4–3
b. log 3 81  log 3 27
4
= log3 3 – log3 3
=
4 – 3
3
These examples suggest the following property:
Quotient Property of Logarithms:
For all positive numbers m, n and b where b ≠ 1,
logb m = logb m – logb n
n
Examples: Given log12 9 = 0.884 and
log12 18 = 1.163, find each:
a.
3 
= log12 9
log 12  
4 
12
= log12 9 – log12 12
= 0.884 – 1
= –0.116
= log12 18
b. log12 2
9
= log12 18 – log12 9
= 1.163 – 0.884
= 0.279
Consider the following:
4
Evaluate
a. log3 9
2 4
= log3 (3 )
2•4
= log3 3
=
2•4
b. 4 log3 9
2
= (log3 3 ) • 4
=
2
•4
These examples suggest the following property:
Power Property of Logarithms:
For all positive numbers m, n and b where b ≠ 1,
p
logb m = p • logb m
Example: Expand log10 7x3
3
log10 7 + log10 x
log10 7 + 3log10 x
Example: Expand log2 85/3x2y4
2
5/3
log2 8 + log2 x + log2 y4
5 log2 8 + 2log2 x + 4log2 y
3
Example: Expand
loga 4xy
3
z
2
This logarithm contains several operations that can be
expanded…
Multiplication expands to addition;
The exponent expands to multiplication;
Division expands to subtraction…
loga 4 + loga x + 2 loga y – 3 loga z
Assignment
p. 416: 5 – 24 all