How does $\exp(x+y) = \exp(x)\exp(y)$ imply $\exp(x) = [\exp(1)]^x$? In Calculus by Spivak (1994), the author states in Chapter 18 p. 341 that 
$$\exp(x+y) = \exp(x)\exp(y)$$ implies 
$$\exp(x) = [\exp(1)]^x$$ 
He refers to the discussion in the beginning of the chapter where we define a function $f(x + y) = f(x)f(y)$; with $f(1) = 10$, it follows that $f(x) = [f(1)]^x$. But I don't get this either. Can anyone please explain this? Many thanks!

$$
\begin{gathered}
  \exp (x + y) = \exp (x)\exp (y)\quad  \Rightarrow  \hfill \\
   \Rightarrow \quad \exp \left( {\sum\limits_j {x_j } } \right) = \prod\limits_j {\exp (x_j )} \quad  \Rightarrow  \hfill \\
   \Rightarrow \quad \exp \left( {\sum\nolimits_{k\, = \,0}^{\,x} 1 } \right) = \prod\nolimits_{k\, = \,0}^{\,x} {\exp (1)} \quad  \Rightarrow  \hfill \\
   \Rightarrow \quad \exp \left( x \right) = \exp (1)^{\,x}  \hfill \\ 
\end{gathered} 
$$
where 


*

*$\sum\limits_j {} $ and $\prod\limits_j {} $ indicate the sum and product over a set of values for $j$

*$\sum\nolimits_{k\, = \,0}^{\,x} {} $ and $\sum\nolimits_{k\, = \,0}^{\,x} {} $ indicate the Indefinite sum/product, computed between the given limits 


So the above holds also for $x$ and $y$ complex
