Power of Positive Real Number is Positive/Real Number

Theorem

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

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

Then:

$x^r > 0$

where $x^r$ denotes the $x$ to the power of $r$.

Proof

From the definition of $x$ to the power of $r$:

$x^r = \map \exp {r \ln x}$

The result follows from Exponential of Real Number is Strictly Positive.

$\blacksquare$