(related to Proposition: Exponential Function of General Base With Integer Exponents)
It has been shown already, that for positive real numbers \(a > 0\) and an integer \(n\ge 0\), the n-th-power function equals the exponential function of general base with natural exponents: \[a^n=\exp_a(n).\quad\quad( * )\] Let \(n < 0\). From the reciprocity law, it follows \[a^{-n}=\frac{1}{a^n}=\frac{1}{\exp_a(n)}=\exp_a(-n).\] Thus, \( ( * )\) is valid for all integers \(n\in\mathbb Z\).
