# What is the domain of the natural logarithm?

Mar 21, 2016

See explanation...

#### Explanation:

A natural logarithm is the inverse of the Real exponential function $x \to {e}^{x}$ or the Complex exponential function $z \to {e}^{z}$.

The domain of ${e}^{x}$ is the whole of $\mathbb{R}$, while its range is $\left(0 , \infty\right)$. The exponential function is one-one on its domain, so its inverse - the Real natural logarithm, $\ln x$ - is well defined, with domain $\left(0 , \infty\right)$ and range $\mathbb{R}$.

The Complex logarithm is somewhat messier. $z \to {e}^{z}$ is many to one since ${e}^{2 \pi i} = 1$. So we need to restrict the domain of ${e}^{z}$ or the range of $\ln z$ in order to be able to define $\ln z$ as a function.

We can define the principal natural logarithm of a Complex number as follows:

$\ln z = \ln \left\mid z \right\mid + \text{Arg} \left(z\right) i$

The definition of (the principal value of) $\text{Arg} \left(z\right)$ may be in the range $\left(- \pi , \pi\right]$ or $\left[0 , 2 \pi\right)$, resulting in different possible definitions for the principal Complex logarithm.

The one number for which we cannot define a meaningful logarithm is $0$.