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
This follow directly from the fact that the exponential function is the inverse of the natural logarithm function. According to the definition for exponent, $e^{x}=y\iff\ln y=x$. Raising the equation $\ln y=x$ to the power of $e$ we receive $e^{\ln y}=e^{x}\implies e^{\ln y}=y$