Usually the question about Iwasawa logarithm is why $\log_p p = 0$, but here I am interested in justifying why $\log_p$ must be null over the group of roots of 1 of order prime to p.
The reason I have seen given is that if $\zeta^n = 1$ then $n\cdot\log_p\zeta = \log_p 1 = 0$, but one could actually argue the same for $e^{i\frac{p}{q}(2\pi)}$ (with $p/q$ any rational), yet no definition of complex logarithm implies it to be null over the unit circle $\mathbb{U}$ (at least as far as I know).
Are there other reasons ?