Exponential of Product

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

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

Then:
 * $\exp \left({x y}\right) = \left({\exp y}\right)^x$

Proof 1
Let $Y = \exp y$.

From Exponential of Natural Logarithm:
 * $\ln \left({\exp y}\right) = y$

From Logarithms of Powers, we have:
 * $\ln Y^x = x \ln Y = x \ln \left({\exp y}\right) = x y$

Thus:
 * $\exp \left({x y}\right) = \exp \left({\ln Y^x}\right) = Y^x = \left({\exp y}\right)^x$