I've been working with logarithms of negative numbers and I've realized that most of the common log rules don't hold for $x<0$. While this is stated in all the proofs I can find, I can't seem to work out exactly why that assumption is necessary, and where the proof fails without it. Does anyone know a good example of where and why a common log rule proof fails without this assumption?
Ex (power rule):
So the power rule has failed in this case. Why?