# Definition:Exponential Function/Real/Inverse of Natural Logarithm

## Definition

Consider the natural logarithm $\ln x$, which is defined on the open interval $\left({0 \,.\,.\, +\infty}\right)$.

From Logarithm is Strictly Increasing:

- $\ln x$ is strictly increasing.

From Inverse of Strictly Monotone Function:

- the inverse of $\ln x$ always exists.

The inverse of the natural logarithm function is called the **exponential function**, which is denoted as $\exp$.

Thus for $x \in \R$, we have:

- $y = \exp x \iff x = \ln y$

The number $\exp x$ is called the **exponential of $x$**.

The domain of $\exp$ is $\R$, and the codomain of $\exp$ is $\R_{>0}$.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $48$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 14.4$

- Weisstein, Eric W. "Exponential Function." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialFunction.html