4.4A Properties of Logarithms
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equation or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context.
F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
Bell Work 4.4:
Simplify.
1. (26)(28)
2. (3-2)(35)
Write in exponential form.
6. logx x = 1
3.
312
34
4.
43
4 -1
5. (73)5
7. 0 = logx 1
Anticipatory Set:
Since logarithms are exponents, the laws of exponents apply to them.
Product Property of Logarithms
For any positive numbers m, n, and b (b ≠ 1)
The logarithm of a product is equal to the sum of the logarithms of its factors.
logb mn = logb m + logb n
Example: log 1000 = log (10 · 100) = log 10 + log 100
3
= 1 +
2
This property can be used both forward and backward.
Open the book to page 256 and read example 1.
Example: a. Express log6 4 + log6 9 as a single logarithm. Simplify, if possible.
log6 (4 · 9) = log6 36 = 2
Recall: 62 = 36
b. Express log8 4 + log8 2 as a single logarithm. Simplify, if possible.
log8 (4 · 2) = log8 8 = 1
c. Express log7 (49x) as a sum of two logarithms. Simplify, if possible.
log7 49 + log7 x = 2 + log7 x
d. Express log6 [(x + 2)(x - 1)] as a sum of two logarithms. Simplify, if possible.
log6 (x + 2) + log6 (x – 1)
White Board Activity:
Practice: Express as a single logarithm. Simplify, if possible.
1. log5 5 + log5 25
2. Log1/3 27 + log 1/3 1/9
Log5 (5 ∙ 25)
log1/3 (27 ∙ 1/9)
Log5 125 = 3
log 1/3 3 = -1
1
1
Recall: 5 = 125
Recall:   = 3
3
Practice: Express as a sum of two logarithms. Simplify, if possible.
3. log7 (7xy)
4. log3 (x + 2)(x - 3)
Log7 7 + log7 xy
log3 (x + 2) + log3 (x – 3)
1 + log7 xy
3
Instruction:
Quotient Property of Logarithms
For any positive numbers m, n, and b (b ≠ 1),
The logarithm of a quotient is equal to the difference between the logarithm if the dividend and the
logarithm of the divisor.
logb m/n = logb m - logb n
Example: log2 (16/2) = log2 16 – log2 2
3
=
4 – 1
This property can be used both forward and backward.
Open the book to page 257 and read example 2.
Example: a. Express log5 100 – log5 4 as a single logarithm. Simplify, if possible.
log5 (100/4) = log5 25 = 2
Recall: 52 = 25
b. Express log2 50 - log2 25 as a single logarithm. Simplify, if possible.
log2 (50/25) = log2 2 = 1
c. Express log3 x/9 as a difference of two logarithms. Simplify, if possible.
log3 x – log3 9 = log3 x – 2
d. Express log5 y/(1/5) as a difference of two logarithms. Simplify, if possible.
log5 y – log5 (1/5) = log5 y – (-1) = log5 y + 1
White Board Activity:
Practice: a. Express log7 49 – log7 7 as a single logarithm. Simplify, if possible.
log7 49/7 = log7 7 = 1
Recall: 71 = 7
b. Express log2 64 - log2 8 as a single logarithm. Simplify, if possible.
Log2 64/8 = log2 8 = 3
c. Express log3 (27/3) as a difference of two logarithms. Simplify, if possible.
Log3 27 – log3 3 = 3 – 1 = 2
d. Express log (100/4x) as a difference of two logarithms. Simplify, if possible.
Log 100 – log 4x = 2 – log 4x
Power Property of Logarithms
For any number p and positive numbers a and b (b ≠ 1),
The logarithm of a power is the product of the exponent and the logarithm.
logb ap = p logb a
log 103 = log (10 · 10 · 10) = log 10 + log 10 + log 10 = 3 log 10
log 1000 = 3 log 10
3
=3·1
Open the book to page 257 and read example 3.
Example: Express as a product. Simplify, if possible.
a. log2 326 = 6 log2 32 = 6 · 5 = 30
b. log8 420 = 20 log8 4
White Board Activity:
Practice: Express as a product. Simplify if possible.
1. log 104
2. Log5 252
4 log 10
2 log5 25
4·1=4
2·2=4
Recall: 25 = 32
3. Log2 (1/2)5
5 log2 (1/2)
5 · -1 = -5
Complete Practice 4.4A
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 260 – 261 prob. 1 – 10, 20 – 28, 40, 41, 43.
For a Grade:
Text: pgs. 260 – 261 prob. 2, 4, 8, 20, 24, 26.