Considering $\mathbb{C}_p$, a completion of an algebraic closure of the field of $p$-adic numbers $\mathbb{Q}_p$, it is well known that it is possible to define a 'logarithm' function:
$$\log_p : \mathbb{C}_p^\times \longrightarrow \mathbb{C}_p$$
s.t. $\log_p(xy) = \log_p(x) + \log_p(y)$, $\log_p(p)=0$ and if $|x|<1$ then $\log_px$ is given by the usual power series. this function is called Iwasawa logarithm.
If we denote by $\log$ the classical logarithm given by the power series, we know in the $p$-adic setup it converges, in fact, for $|x| < 1$.
In general if $x \in \mathbb{C}_p^\times$ then $x = p^\alpha \zeta u$ with $\zeta$ root of 1 and $|u| =1$, so thank to the properties of the Iwasawa logarithm it's easy to see:
$$\log_px = \log \left( \frac{x}{p^\alpha \zeta} \right)$$
Moreover it is known that (denote $\mathbb{M}_p$ the maximal ideal of $\mathbb{C}_p$):
$$ \log_p \left(1+ \mathbb{M}_p\right) = \log \left( 1+\mathbb{M}_p\right) = \mathbb{C}_p $$
in particular the logarithm is not bounded. Is it possible to give estimates to $|\log_px|$ by using $|x|$, or better, to calculate the norm of $\log_px$.
The most famous 'bound' for $\log$ is:
$$ \log( 1 + D(0, r_p)) = D(0, r_p) $$ where $r_p$ is the radius of convergence of the $p$-adic exponential.
Assume to take now $x,y \in \mathbb{C}_p^\times$:
$$ \log_p(x+y) = \log_px + \log_p\left( 1 + \frac{y}{x}\right) $$ asking $|y/x|< r_p$ we get $|\log_p\left( 1 + \frac{x}{y}\right)|<r_p$ but we have no control for $\log_px$ because in general $|\frac{x}{p^\alpha \zeta}|=1$
Are there known results for giving bounds to the Iwasawa logarithm? In genenral I'm also interested if there are other points of $\mathbb{C}_p^\times$ that are mapped inside $D(0, r_p)$ except for $D(1, r_p)$