In a problem I am working on, I am making the simplification $$\frac{\pi i+\log(a)}{\log(b)} = \log_b(-a).$$ Here, $a,b$ are complex numbers with $|b|>1$ and $a \neq 0,-1$ and $ab^n \neq-1$ for any non-negative integer $n$. By making the above simplification, what assumptions am I making about $a,b$? That is, what new restrictions does the above statement introduce?
I think I am only assuming that if $a,b$ are real numbers, then they are both positive. Is there anything else?