Properties of Logarithms
Logarithms have the following properties:
Since a^{0} = 1 and a^{1} = a:
Property A: log_{a}1 = 0
Property B: log_{a}a = 1
Since a^{x} and log_{a}x are inverses:
Property C: log_{a}a^{x} = x
Property D: a^{logax} = x
Since a^{p}a^{q} = a^{p+q} and = a^{pq}:
Property E: log_{a}(pq) = log_{a}p + log_{a}q
Property F: log_{a}() = log_{a}p  log_{a}q
Since log_{a}(M^{n}) = log_{a}(M·M·M^{ ... }M) = log_{a}M + log_{a}M + log_{a}M + ^{ ... } + log_{a}M = n·log_{a}M
Property G: log_{a}(M^{n}) = n·log_{a}M
Property H
Logarithms have an additional property, called property H, and a property H_{1} that is a specific case of property H.
Property H: log_{a}M = , where b is any base.
Property H_{1}: log_{a}M =
Applications of Properties
The numerous properties listed on this page can be used to evaluate logarithmic functions. Property
H_{1} is especially useful when evaluating logarithms with a calculator:
since most calculators only evaluate logarithms with base 10, we can evaluate
log_{a}M by evaluating . For example, log_{3}4 = .
Example:
log_{5}10 + log_{5}20  log_{5}8 =?
 =  log_{5}() 

 =  log_{5}25 

 =  log_{5}5^{2} 

 =  2. 
