Source: Spivak's Calculus Chapter 18: The Logarithm and Exponential Functions. Theorem 3:
Theorem: For all numbers $x$, $\exp(x+y)=\exp(x)\exp(y)$, where $\exp$ is defined as $\log^{-1}$
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Proof: Let $x' = \exp(x)$ and $y' = \exp(y)$, so that
$x = \log x'$,
$y = \log y'$.
Then
$x + y = \log x' + \log y' = \log(x'y')$.
This means that
$\exp(x + y) = x'y' = \exp(x) \exp(y)$.
I don't get the beginning part where he lets $x' = \exp(x)$ and $y'=\exp(y)$... Could he have used $f'= \exp(x)$ and $g'=\exp(y)$ for less confusion, or am I misunderstanding something completely?