why does e^2x = 1/2 e^2x
does e^-x = -e^-x or -1/e^-x ?
i guess, but mathematicians don't take anything for granted, and therefore, they are in the habit of rigorously defining common sense things.
for instance, to define a rational number, a mathematician would say:
a rational number $\displaystyle r$ is one that can be expressed as $\displaystyle r = \frac {a}{b} \mbox { for } a,b \in \mathbb {Z}, b \neq 0$
even though the $\displaystyle b \neq 0$ is the "common sense" part