Definition:General Logarithm

Definition

The natural logarithm function gives rise to the exponential function as follows:

$x = \ln y \iff y = \exp x = e^x$

Thus the logarithm is the inverse of the exponential. It can also be independently shown that the logarithm function always exists without taking recourse to the fact that it is the inverse. For a proof see Existence of Logarithm.

Consider the general exponential function: $y = a^x = e^{x \ln a}$, where $a \in \R: a > 0$, as defined in Powers of Real Numbers.

As $\forall x \in \R: x \ln a \in \R$, and the nature of the exponential function (strictly increasing), we can define the function $\log_a y$:

$x = \log_a y \iff y = a^x$

This is called the logarithm to the base $a$, or log base $a$.

When $a = e$, they are of course natural logarithms, and are sometimes called Napierian logarithms although this name too is dying out.

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

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

Common Logarithms

When $a = 10$, the logarithms are common logarithms, sometimes called Briggsian Logarithms. Before the advent of cheap means of electronic calculation, they used to be important.

In elementary textbooks and on most pocket calculators, $\log$ is assumed to mean $\log_{10}$. This ambiguous notation is not recommended, particularly since $\log$ often means base $e$ in more advanced textbooks.

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.2$