$x^a$ not defined in $\Re$ when $x$ negative?

Question

I know that by definition $x^a = \exp(a\log x)$ which is not defined when $x$ not in $\Re^∗_+$

Does it mean that for example $(-1)^3$ not defined for real numbers?

Answer 1

No, $(-1)^3= (-1)(-1)(-1)= -1$ is perfectly well defined. Your formula, $x^a= e^{a log(x)}$, is only correct when that logarithm exists.

Answer 2

$(-1)^a$ is only real if $a$ is a rational number with an odd denominator. $$(-1)^a = \left\{ \begin{array}{rl} 1 & a=0 \\ -1& a\in\{ \frac{p}{2q+1},p\in Z_+ , q\in Z \} \\ undefined & otherwise\\ \end{array} \right.$$

which is a pretty bizarre discontinuous function

so $(-1)^3 = -1$

but $\lim_{x \to3}(-1)^x$ does not exist.