Exponent of Sum/Real Numbers/Corollary

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Corollary to Exponent of Sum: Real Numbers

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

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


Then:

$\exp \left({x - y}\right) = \dfrac {\exp x} {\exp y}$


Proof

By Exponent of Sum: Real Numbers:

$\exp \left({x - y}\right) = \exp x \exp \left({- y}\right)$


By Reciprocal of Complex Exponential:

$\dfrac 1 {\exp y} = \exp \left({- y}\right)$

Combining these two, we obtain the result:

$\exp \left({x - y}\right) = \dfrac {\exp x} {\exp y}$

$\blacksquare$