Example

Suppose that K is locally compact. Let U be the set of invertible elements of A; it is a compact open subgroup of the Lie group $K^*$. Then $U \subset (K^*)_f$ by Proposition 13; on the other hand, if $x \in K^*$ is such that $x \notin U$, either $x^n$ tends to 0 as n tends to $+\infty$, or $x^n$ tends to 0 as n tends to $-\infty$; hence $U = (K^*)_f$. The function $\log_{K^*}$ is defined and analytic on U, with values in $L(K^*) = K$, and such that $\log_{K^*}(xy) = \log_{K^*}(x) + \log_{K^*}(y)$ for all $x, y$ in U; the elements x of U such that $\log_{K^*}(x) = 0$ are the roots of unity of K.

We again use the notation of nos. 3, 4 and 5.