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
Let $$Y = \exp y$$.

From the definition of the Basic Properties of Exponential Function, $$\ln \left({\exp y}\right) = y$$.

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

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