Exponential functions with negative base

Question

Consider the function $f(x) = (-2)^x$, $x$ belongs to irrationals. For which $x$ does $f(x)$ belong to the reals.

Answer 1

Rewrite

$$ f(x) = (-2)^x = 2^x \mathbf{i}^{2x} = 2^x \Big( \cos(\pi x) + \mathbf{i} \sin(\pi x ) \Big). $$

So real for

$$ \sin(\pi x) = 0 \quad \Longrightarrow \quad x \in \mathbb{Z}. $$

General base

$$ f(x) = (-b)^x =b^x (-1)^x = b^x \mathbf{i}^{2x} = b^x \Big( \cos(\pi x) + \mathbf{i} \sin(\pi x ) \Big). $$

So real for

$$ \sin(\pi x) = 0 \quad \Longrightarrow \quad x \in \mathbb{Z}. $$