Exponential of Natural Logarithm

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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 $e$:

\(\displaystyle e^{\ln x}\) \(=\) \(\displaystyle e^y\)
\(\displaystyle \) \(=\) \(\displaystyle x\)

$\blacksquare$