calculus - 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!

Answer

I think you should assume $x$ is an integer (since $a^x$ is defined using $\exp$ if $x$ is a positive real). You can write $\exp(x) = \exp(\underbrace{1+1+1+1+\cdots+1}_x)$.

Using the property of $\exp$, you find that $\exp(x) = \exp(1)\exp(1)\dots\exp(1) = (\exp(1))^x$.