Exponential of Sum of Real Numbers/Corollary

Corollary to Exponential of Sum of Real Numbers

Let $x, y \in \R$ be real numbers.

Let $\exp x$ be the exponential of $x$.

Then:

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

Proof

By Exponential of Sum of Real Numbers:

$\map \exp {x - y} = \exp x \, \map \exp {-y}$

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By Reciprocal of Complex Exponential:

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

Combining these two, we obtain the result:

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

$\blacksquare$