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.
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. log b (MN)  log b M  log b N
M 
2. log b   log b M  log b N
 N 
3. log b M p  plog b M
Example
• Use logarithmic properties to expand the
expression as much as possible.
2
5x
2
log 2
 log 2 5 x  log 2 3
3
Example cont.
2
5x
2
log 2
 log 2 5 x  log 2 3
3
2
 log 2 5  log 2 x  log 2 3
Example cont.
2
5x
2
log 2
 log 2 5 x  log 2 3
3
2
 log 2 5  log 2 x  log 2 3
 log 2 5  2 log 2 x  log 2 3
Properties for Condensing
Logarithmic Expressions
• For M > 0 and N > 0:
1. log b M  log b N  log b (MN)
M 
2. log b M  log b N  log b  
N 
3. plog b M  log b M
p
The Change-of-Base Property
• For any logarithmic bases a and b, and any
positive number M,
log a M
log b 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
log 10 3
ln 7
log 3 7 
ln 3
log 3 7  1.77
Properties of
Logarithms