(related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
By virtue of the functional equation of the exponential function we have \[\exp(x)\cdot \exp(-x)=\exp(x-x)=\exp(0).\quad\quad( * )\] From the result that \[\exp(0)=1\] it follows from the uniqueness of reciprocal numbers that \[\exp(-x)=\exp(x)^{-1}.\] In particular, \(\exp(x)\neq 0\) for all \(x\in\mathbb R\).1
\(\exp(x)=0\) for some \(x\) would imply \(0\cdot\exp(-x)=1\), which cannot be fulfilled for any real number \(\exp(-x)\). ↩
