It is well known that we can define the logarithm in the p-adic setup by the usual power series that converges in $$B(1,1^-)$$. In Schickoff "Ultrametric calculus" there is an extension of the $$log$$ from the unit ball to $$C_p^{\times}$$ (called $$LOG$$) and it is proved that it is locally analytic, then he defines the Iwasawa Logarithm as the function $$LOG(x) -ord_p(x)$$ and this is the unique multiplicative function extending the logarithm of the unit ball such that vanishes on $$p$$. My question is, this function is still locally analytic on $$C_p^{\times}$$?

• Perhaps you should recall precisely the definition of LOG – nguyen quang do Nov 2 '18 at 15:34

Sure. Local analyticity can be judged solely from the behavior at the identity, $$1$$. And there, the function is given by the series that you know.