Definition:Natural Logarithm/Positive Real/Definition 1

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The (natural) logarithm of $x$ is the real-valued function defined on $\R_{>0}$ as:

$\ds \forall x \in \R_{>0}: \ln x := \int_1^x \frac {\d t} t$


The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as:

$\ln z$
$\log z$
$\Log z$
$\log_e z$

The first of these is commonly encountered, and is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated.

The second and third are ambiguous (it doesn't tell you which base it is the logarithm of).

While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.

Also see

  • Results about logarithms can be found here.

Historical Note

The natural logarithm was discovered by accident by John Napier in around $1590$, evolving from his invention of the Napierian logarithm as a tool for multiplication of numbers by addition.

He had no concept of the notion of the base of a logarithm and certainly did not use Euler's number $e$.