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Properties of
Logarithms
The Product Rule
• Let b, M, and N be positive real numbers
with b ≠ 1.
• logb (MN) = logb M + logb N
• The logarithm of a product is the sum of the
logarithms.
• For example, we can use the product rule to
expand ln (4x): ln (4x) = ln 4 + ln x.
The Quotient Rule
• Let b, M and N be positive real numbers
with b ≠ 1.
M
log b   = log b M − lobb N
N
• The logarithm of a quotient is the difference
of the logarithms.
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The Power Rule
• Let b, M, and N be positive real numbers
with b = 1, and let p be any real number.
• log b M p = p log b M
• The logarithm of a number with an
exponent is the product of the exponent and
the logarithm of that number.
Text Example
Write as a single logarithm:
a. log4 2 + log4 32
Solution
a. log4 2 + log4 32 = log4 (2 • 32)
= log4 64
=3
Use the product rule.
Although we have a single logarithm,
we can simplify since 43 = 64.
Properties for Expanding
Logarithmic Expressions
• For M > 0 and N > 0:
1. logb (MN) = log b M + log b N
 M
2. logb   = logb M − log b N
 N
3. log b M p = plog b M
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Example
• Use logarithmic properties to expand the
expression as much as possible.
5x 2
log 2
= log 2 5 x 2 − log 2 3
3
Example cont.
5x 2
= log 2 5 x 2 − log 2 3
log 2
3
= log 2 5 + log 2 x 2 − log 2 3
Example cont.
5x 2
= log 2 5 x 2 − log 2 3
log 2
3
= log 2 5 + log 2 x 2 − log 2 3
= log 2 5 + 2 log 2 x − log 2 3
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Properties for Condensing
Logarithmic Expressions
• For M > 0 and N > 0:
1. logb M + log b N = log b (MN)
M
2. logb M − log b N = log b  
N
3. plog b M = logb M p
The Change-of-Base Property
• For any logarithmic bases a and b, and any
positive number M,
log b M =
log a M
log a b
• The logarithm of M with base b is equal to the
logarithm of M with any new base divided by the
logarithm of b with that new base.
Example
Use logarithms to evaluate log37.
Solution:
log 7
log 3 7 =
or
so
10
log10 3
log 3 7 =
ln 7
ln 3
log 3 7 = 1.77
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Properties of
Logarithms
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