1. 8.3 Properties of logarithms
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2. Find the inverse of each: 1. g(x) = 5x 2. f(x) = 2 + log4x
Warm-up
Find the inverse of each:
1. g(x) = 5x f(x) = 2 + log4x
Simplify each:
x = 5y
y = log5 x
x = 2 + log4 y
x – 2 = log4 y
y = 4x – 2
x 7
x 15
x 6

3. 67.
68.
69.
70.
59. g(x) = 6x
60. g(x) = log8x
61. g(x) = log1/3 x
62. g(x) = (1/2)x

4. Properties of Logarithms
Consider the following two problems:
Simplify log3 (9 • 27)
= log3 (32• 33)
= log3 (32 + 3) =
Simplify log3 9 + log3 27
= log log3 33 =
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

5. We will use the Product Property of Logarithms to solve problems...
Example Given log2 5 = 2.322, find log2 40
log2 20 = log2 (23 • 5)
= log log2 5 = =

6. Consider the following: a. b.
= log
= log3 34 – 3
= – 3
= log3 34 – log3 33
= – 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

7. = log
= log12 9 – log12 12
= – 1
= –0.116
= log
= log12 18 – log12 9
= – 0.884 =
Examples: Given log12 9 = and log12 18 = 1.163, find each: a b. log12 2

8. Consider the following: Evaluate a. log3 94 b. 4 log3 9
= • 4
= (log3 32) • 4
= • 4
These examples suggest the following property:
Power Property of Logarithms: For all positive numbers m, n and b where b ≠ 1, logb mp = p • logb m
Consider the following: Evaluate a. log b. 4 log3 9

9. Example: Expand log10 7x3
log log10 x3
log log10 x
Example: Expand log2 85/3x2y4
log2 85/3 + log2 x2 + log2 y4
5 log log2 x + 4log2 y 3

10. Example: Expand loga 4xy2 z3
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

11. Assignment
p. 416: 5 – 24 all