# Reciprocal of Real Exponential

## Theorem

Let $x \in \R$.

Let $\exp$ denotes the real exponential function.

Then:

$\dfrac 1 {\map \exp x} = \map \exp {-x}$

## Proof

 $\displaystyle \map \exp 0$ $=$ $\displaystyle 1$ Exponential of Zero $\displaystyle \leadsto \ \$ $\displaystyle \map \exp {x - x}$ $=$ $\displaystyle 1$ as $\forall x \in \R : x - x = 0$ $\displaystyle \leadsto \ \$ $\displaystyle \map \exp {-x} \, \map \exp x$ $=$ $\displaystyle 1$ Exponential of Sum: Real Numbers $\displaystyle \leadsto \ \$ $\displaystyle \map \exp {-x}$ $=$ $\displaystyle \dfrac 1 {\exp x}$ dividing both sides by $\exp x$, which is non-zero by Exponential of Real Number is Strictly Positive

$\blacksquare$