The logarithm for a base and a number is defined to be the inverse function of taking to the power , i.e., . Therefore, for any and ,
(1)

or equivalently,
(2)

For any base, the logarithm function has a singularity at . In the above plot, the blue curve is the logarithm to base 2 (), the black curve is the logarithm to base (the natural logarithm ), and the red curve is the logarithm to base 10 (the common logarithm, i.e., ).
Note that while logarithm base 10 is denoted in this work, on calculators, and in elementary algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use the notation to mean , and therefore use to mean the common logarithm. Extreme care is therefore needed when consulting the literature.
The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation to denote the nested natural logarithm .
In the Wolfram Language, the logarithm to the base is implemented as Log[b, x], while Log[x] gives the natural logarithm, i.e., Log[E, x], where E is the Wolfram Language symbol for e.
Whereas powers of trigonometric functions are denoted using notations like , is less commonly used in favor of the notation .
Logarithms are used in many areas of science and engineering in which quantities vary over a large range. For example, the decibel scale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightnesses are all logarithmic scales.
The derivative and indefinite integral of are given by
(3)
 
(4)

The logarithm can also be defined for complex arguments, as shown above. If the logarithm is taken as the forward function, the function taking the base to a given power is then called the antilogarithm.
For , is called the characteristic, and is called the mantissa.
Division and multiplication identities for the logarithm can be derived from the identity
(5)

Specifically, for ,
(6)
 
(7)
 
(8)

There are a number of properties which can be used to change from one logarithm base to another, including
(9)
 
(10)
 
(11)
 
(12)
 
(13)
 
(14)
 
(15)
 
(16)
 
(17)

An interesting property of logarithms follows from looking for a number such that
(18)

(19)

(20)

(21)

so
(22)

Another related identity that holds for arbitrary is given by
(23)

Numbers of the form are irrational if and are integers, one of which has a prime factor which the other lacks. A. Baker made a major step forward in transcendental number theory by proving the transcendence of sums of numbers of the form for and algebraic numbers.