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:

$\forall x > 0: \map \exp {\ln x} = x$
$\forall x \in \R: \map \ln {\exp x} = x$


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 $e$:

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

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.