(related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
By definition, we have \[\exp_a(0)=\exp(0\cdot\ln(a))=\exp(0)=1\] by the corresponding proposition. The functional equation of the exponential function of general base gives us \[\exp_a(x+y)=\exp_a(x)\cdot \exp_a(x)\] for all real numbers \(x,y,a\) with \(a > 0\). If we set \(y=-x\), we get \[1=\exp_a(x-x)=\exp_a(x)\cdot \exp_a(-x)\] This implies with the uniqueness of reciprocal numbers \[\exp_a(-x)=\frac 1{\exp_a(x)}.\] In particular, we have \[\exp_a(x)\neq 0\] for all \(x\in\mathbb R\).
