Exponential of Natural Logarithm

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:
 * $\forall x > 0: \exp \left({\ln x}\right) = x$
 * $\forall x \in \R: \ln \left({\exp x}\right) = x$

Proof
From the definition of the exponential function:
 * $e^y = x \iff \ln x = y$

Raising both sides of the equation $\ln x = y$ to the power of $e$: