Is there a problem when defining exponential with negative base?

It depends what number system you are working in. In the reals, there is no answer. In the complex numbers, there are many answers. You can take the principal branch of the logarithm function, just saying that the imaginary part of the result will be in the range $(-\pi,\pi]$, in which case it is well defined. But if $e^z=w$, it is also true that $e^{z+2i\pi k}=w$ for $k \in \Bbb Z$ so you could say $\log w=z + 2i \pi k$