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?