Exponential of Real Number is Strictly Positive/Proof 4

Theorem

Let $x$ be a real number.

Let $\exp$ denote the (real) exponential function.

Then:

$\forall x \in \R : \exp x > 0$

Proof

This proof assumes the definition of $\exp$ as the inverse mapping of extension of $\ln$, where $\ln$ denotes the natural logarithm.

Recall that the domain of $\ln$ is $\R_{>0}$.

From the definition of inverse mapping, the image of $\exp$ is the domain of $\ln$.

That is, the image of $\exp$ is $\R_{>0}$.

Hence the result.

$\blacksquare$