Exponential of Natural Logarithm

Theorem

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

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

Let $\ln x$ be the natural logarithm 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 $e$:

\(\ds e^{\ln x}\)

\(=\)

\(\ds e^y\)

\(\ds \)

\(=\)

\(\ds x\)

$\blacksquare$