Exponential of Product/Proof 2

Theorem

Let $x, y \in \R$ be real numbers.

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

Then:

$\map \exp {x y} = \paren {\exp y}^x$

Proof

\(\ds \paren {\exp y}^x\)

\(=\)

\(\ds \map \exp {x \map \ln {\exp y} }\)

Definition of Power to Real Number

\(\ds \)

\(=\)

\(\ds \map \exp {x y}\)

Exponential of Natural Logarithm

$\blacksquare$