Definition:Natural Logarithm

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Definition

Positive Real Numbers

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

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


Complex Numbers

Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.

The complex natural logarithm of $z \in \C_{\ne 0}$ is the multifunction defined as:

$\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$

where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.


Notation

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.


Examples

Natural Logarithm: $\ln 2$

Mercator's constant is the real number:

\(\ds \ln 2\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n\)
\(\ds \) \(=\) \(\ds 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb\)
\(\ds \) \(=\) \(\ds 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots \ldots\)


Natural Logarithm: $\ln 3$

The natural logarithm of $3$ is:

$\ln 3 = 1.09861 \, 22886 \, 68109 \, 69139 \, 5245 \ldots$


Natural Logarithm: $\ln 10$

The natural logarithm of $10$ is approximately:

$\ln 10 \approx 2 \cdotp 30258 \, 50929 \, 94045 \, 68401 \, 7991 \ldots$


Also known as

The natural logarithm is sometimes referred to as the Napierian logarithm for John Napier, although this was not actually the logarithm he was famous for inventing.


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 $e$.


Sources

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