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In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.

For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000. As another example, the logarithm of 32 to the base 2 is 5, because 2 raised to the power 5 is 32.

The logarithm of $x$ to the base $b$ is written $\log _{b}\,(x)$ or, if the base is implicit, as $\log(x)$. So, for a number $x$, a base $b$ and an exponent $y$:

${\text{if }}b^{y}=x,{\text{ then }}y=\log _{b}\,(x)$

The $\log _{b}\,(x)$ is a unique real number when $x$ and $b$ are restricted to positive real numbers and is negative for $0<b<1$, zero for $b=1$, and positive for $b>1$.

Features of the logarithm

An important feature of logarithms is that they reduce multiplication to addition. That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers as in this identity:

$\log(xy)=\log x+\log y$

Logarithms also reduce division to subtraction as in this identity:

$\log(x/y)=\log(x)-\log(y)$

And they reduce exponation to multiplication as in this identity:

$\log(x^{y})=y\log(x)$

And taking roots are reduced to division:

$\log({\sqrt[{y}]{x}})={\frac {\log(x)}{y}}$

The inverse of the logarithm is call the antilogarithm and it is expressed in this identity:

${\text{antilog}}_{b}(n)=b^{n}$

Although the above practical advantages are not important for
 numerical work today, they are used in graphical analysis (see Bode plot).