Exponent Combination Laws/Product of Powers/Proof 1

Theorem

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

Let $x, y \in \R$ be real numbers.

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

Then:

$a^x a^y = a^{x + y}$

Proof

\(\ds a^{x + y}\)

\(=\)

\(\ds \map \exp {\paren {x + y} \ln a}\)

Definition of Power to Real Number

\(\ds \)

\(=\)

\(\ds \map \exp {x \ln a + y \ln a}\)

\(\ds \)

\(=\)

\(\ds \map \exp {x \ln a} \, \map \exp {y \ln a}\)

Exponential of Sum

\(\ds \)

\(=\)

\(\ds a^x a^y\)

Definition of Power to Real Number

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (1) \ \text{(i)}$