The definition of Euler's constant to the power $x$, $e^x$, is

$$e^x = \sum\_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + {...}$$

And of course, we have the number $e$ defined as

$$e = \sum\_{n=0}^\infty \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + {...}$$
or
$$e = \lim\_{n\to \infty} (1+\frac{1}{n})^{n}$$

$e$ and $e^x$ here are expressions of a sum of infinite series. When one calculate $e^x$, he doesn't go by the definition of $e^x$, but instead calculates the *numerical value* of $e$, and *takes the power of that numerical value* directly.

**How can one simply take the power of the *numerical value* of $e$ directly, and be sure the answer is $e^x$? And what about in the context of arbitrary powers of $e$?**

p.s There are also different definitions of $e$, like:
$$\int\_1^{e^x}{\frac{1}{t}dt}=x$$
$$\frac{d}{dx}e^x = e^x$$
$$\frac{d}{dx}log\_e{x}=\frac{1}{x}$$
But they do not explain the concern too.