Logarithms have the following properties:
Since
a
^{0} = 1
and
a
^{1} = a
:
Property A: log_{a}1 = 0Since a ^{x} and log_{a} x are inverses:
Property B: log_{a} a = 1
Property C: log_{a} a ^{x} = xSince a ^{p} a ^{q} = a ^{p+q} and = a ^{p-q} :
Property D: a ^{logax} = x
Property E: log_{a}(pq) = log_{a} p + log_{a} qSince 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 F: log_{a}() = log_{a} p - log_{a} q
Property G: log_{a}(M ^{n}) = n·log_{a} M
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 =
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. |