Definition:General Logarithm/Positive Real

Definition
Let $x \in \R_{>0}$ be a strictly positive real number.

Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$.

The logarithm to the base $a$ of $x$ is defined as:


 * $\log_a x := y \in \R: a^y = x$

where $a^y = e^{a \ln y}$ as defined in Powers of Real Numbers.

The act of performing the $\log_a$ function is colloquially known as taking logs.

Also known as
The logarithm to the base $a$ of $x$ is usually voiced in the abbreviated form:
 * log base $a$ of $x$

or
 * log $a$ of $x$

When $a = 2$, a notation which is starting to take hold for $\log_2 x$ is $\lg x$. This concept is becoming increasingly important in computer science.

Also see

 * Definition:Real Natural Logarithm: when $a = e$