Exponent Combination Laws/Negative Power

Theorem

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

Let $x \in \R$ be a real number.

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

Then:

$a^{-x} = \dfrac 1 {a^x}$

$\ds a^{-x}$

$=$

$\ds \map \exp {-x \ln a}$

Definition of Power to Real Number

$\ds $

$=$

$\ds \paren {\map \exp {x \ln a} }^{-1}$

$\ds $

$=$

$\ds \frac 1 {\map \exp {x \ln a} }$

$\ds $

$=$

$\ds \frac 1 {a^x}$

Definition of Power to Real Number

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.5$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (1) \ \text{(iii)}$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponent (index)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponent (index)
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.5.$