A typical graph of $f(x) = n^x$ shows only positive solutions:
But it seems like, for some values of $x$, there are negative solutions. For example, $2^{1/2}$ is $\sqrt{2}$, which has a negative solution.
Some say $\sqrt{2}$ is defined to be just the positive root. But I'm not sure this applies to $2^{1/2}$. From the rule $x^a x^b = x^{a+b}$, we get $2^{1/2} = 2^{1/2}$, and can conclude that $2^{1/2} = \pm\sqrt{2}$, with a negative solution.
Is there a reason to define $n^x$ as only having positive solutions?