The logarithm laws are identities that convert products, quotients, and powers into
sums, differences, and multiples
of logarithms. For a real base 
with 
and positive real numbers 
and 
, they are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a real number and the change-of-base formula uses
another valid real base 
.
The third identity is called the log power rule. On the principal branch of the complex logarithm,
product and power identities can differ by integer multiples of 
because of the choice of branch.
