Why can't you have a negative real number for a in an exponential function...

The definition I have says that an exponential function is one that consists of a positive real number, a (not equal to 1), raised to the power of the independent variable, x.

I get why you couldn't have a = 1, but why no negative numbers?

It's because a^x might not be real if a is negative. For instance (-4)^(1/2) = 2i