Category:Natural Logarithms

This category contains results about Natural Logarithms.

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 $r$.