power of an integer

Let $n$ be a non zero integer of absolute value not equal to one. The power of $n$, written $P(n)$ is defined by :

$$P(n)=\frac{\log |n|}{\log \mathrm{rad}(n)}.$$

where $\mathrm{rad}(n)$ is the radical of the integer $n$.

If $n=m^k$, then $P(n)=kP(m)$; in particular, if $n$ is a prime power, $n=p^k$, then $P(n)=k$. This observation explains why the term “power” is usedfor this concept. At the same time, it is worth pointing out that, in general,the power of an integer will not itself be an integer. For instance,

$$P(12)=\frac{\log 12}{\log \mathrm{rad}(12)}=\frac{\log 12}{\log 6}=1.3868\mathrm{\dots }$$

Note that it doesn’t matter what base one uses to compute the logarithm (as long as one uses the same base to compute the logarithm on the numerator and in the denominator!) because, upon changing base, both numerator and denominator will be multiplied by the same factor.