Zeroth Power of Real Number equals One

Theorem

Let $a \in \R_{>0}$ be a (strictly) positive real number.

Let $a^x$ be defined as $a$ to the power of $x$.

Then:

$a^0 = 1$

$\ds a^0$

$=$

$\ds \map \exp {0 \ln a}$

Definition of Power to Real Number

$\ds $

$=$

$\ds \map \exp 0$

$\ds $

$=$

$\ds 1$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.4$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.4.$