Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$ a \mapsto (1+x)^a $$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to extend these characters to locally analytic characters in all $\mathbb{C}_p$?
Thanks for references!
This is a comment, not in any sense an answer.
But you simply can not define $(1+x)^a$ that way, because for $|x|$ very nearly $1$ (in the language of the additive valuation $v(x)=-\log_p(|x|)$: for $v(x)$ very small but positive), $v(\log(1+x))$ will be negative, in other words $\log(x)$ is not even in the ring of integers of the field $\Bbb Q_p(x)$, and certainly not in the domain of any $p$-adic exponential function.
One perfectly satisfactory way of defining $(1+x)^a$ is: $$ (1+x)^a=1+\sum_{k=1}^\infty\binom akx^k\,,\\ \text{where }\binom ak=\frac{a(a-1)\cdots(a-k+1)}{k!}\,. $$ It’s a satisfying exercise to show that if $a\in\Bbb Z_p$, then so is $\binom ak$ for every positive integer $k$.
Another perfectly satisfactory way of defining $(1+x)^a$ is: $$ (1+x)^a=\lim_{|n-a|\to0}(1+x)^n\,, $$ where the values allowed for $n$ are positive integers, and the limit is taken $p$-adically.
Please note: You have written “$\log_p$” for the $p$-adic logarithm, but I have called this simply “log”, while when I wrote “$\log_p$”, I meant the real logarithm to base $p$, as you learn to do in high-school.