How should it be $n$ to apply this property $log(a^n)=nlog(a)$

Question

In a complex analysis text n is suppose to be an integer number, in a calculus text is suppose to be a rational number, the question is, is there actually any restrinction to $n$ to apply the property $log(a^n)=nlog(a)$?

Answer 1

Most general result of this type is the following: (note that powers of complex numbers have to be defined carefully) let $\Omega$ be a region in the complex plane and f be an analytic branch of logarithm on it, i.e.,f is analytic and $e^{f(z)}=z$ for all $z \in \Omega$. Define $a^{z} =e^{zf(a)}$. Then $e^{f(a^{z})}=a^{z}=e^{zf(a)}$ so $f(a^{z})=zf(a)+2k\pi i$. (Note that the integer k is independent of z by continuity). If we write log for f and n for z we got $log(a^{n})=nlog(a)+2k\pi i$. Thus there is no restriction on n but you cannot avoid $2k\pi i$.