(related to Corollary: (Real) Exponential Function Is Always Positive)
For \(x \ge 0\) it follows from the definition of the exponential function that
\[\exp(x)=1+x+\frac {x^2}2+\ldots\ge 1 > 0.\] If \(x < 0\), then \(-x > 0\), and thus \(\exp(-x) > 0\), by the same argument. From the reciprocity of exponential function it follows in this case that \[\exp(x)=\frac 1{\exp(-x)} > 0.\]
Therefore, the exponential function produces only positive real numbers, for all \(x\in\mathbb R\).
