Intermediate Algebra
Section 9.4 – Properties of Logarithms
Since a logarithm is an exponent, there are properties that we have
which are related to properties of exponents that allow us to work
with logarithms.
Properties of Logarithms
Let a, r , and M be positive real numbers such that a ≠ 1 . Then the
following properties are true.
1. log a1 = 0
2. log a a = 1
3. a log a M = M
4. log a a r = r
Example:
a)
Solve the following logarithmic equations.
log 5 54
b)
2log 2 10
Product Property of Logarithms
If M , N , and a are positive real numbers and a ≠ 1 , then
log a MN = log a M + log a N .
Example:
a)
Write the following as a sum of logarithms.
log 5 (7⋅ 4)
b)
log 2 (x ⋅ y )
Section 9.4 – Properties of Logarithms
Example:
a)
page 2
Write each sum as the logarithm of a single expression.
log 3 8 + log 3 4
b)
log 6 3+ log 6 (x + 4) + log 6 5
Quotient Property of Logarithms
If M , N , and a are positive real numbers and a ≠ 1 , then
M
log a
= log a M − log a N .
N
Example:
a)
Write each of the following logarithms as a difference
of logarithms.
 125 
log 5 

 x 
b)
 x −1 
log 2 

 x +1
Section 9.4 – Properties of Logarithms
page 3
The Power Rule of Logarithms
If M and a are positive real numbers and a ≠ 1 , and r is any real
number, then log a M r = r log a M
Example:
a)
log 2 x
Use the Power Rule of Logarithms to express all powers
as factors.
5
Example:
b)
log 5 3 x
Write each expression as a sum or difference of
multiples of logarithms.
2⋅9
13
a)
log 5
c)
log 2 y z
3
b)
log 5
d)
log 3
x
4
y
( x + 5) 2
x
Section 9.4 – Properties of Logarithms
Example:
Write the following as a single logarithm.
a)
3log 5 2
b)
2log 3 5 + log 3 2
c)
1
2log 5 x + log 5 x − 3log 5 (x + 5)
3
d)
5log 6 x −
3
log 6 x + 3log 6 x
4
page 4
Section 9.4 – Properties of Logarithms
page 5
Example: If log b 3 = 0.5 and logb 5 = 0.7 , evaluate the following.
a)
log b
3
5
b)
log b 75
Sometimes we need to evaluate a logarithm with a base that is not
10 or e , and our calculator cannot evaluate logarithms of any other
bases. The following formula allows us to evaluate a logarithm when
the base is neither 10 or e .
Change-of-Base Formula
If a, b, and M are positive real numbers and a ≠ 1 and b ≠ 1 , then
log b M
log a M =
.
log b a
Since our calculator can evaluate common and natural logarithms, we
can always use the Change-of-Base formula with b equal to 10 or e .
Example:
a)
log 4 7
Evaluate the following with the change-of-base
formula.
b)
log 5 8