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We extend to logarithmic class groups the results on abelian principalization of tame ray class groups of a number field obtained in a previous article. As a consequence, for any extension $K/k$ of number fields which satisfies the Gross-Kuz'min conjecture for the prime $\ell$ and where at least one of the infinite places completely splits, we prove that there exists infinitely many abelian $\ell$-extensions $F/k$ such that the relative subgroup $\mathcal{C}\ell_{K/k} = \Ker (\mathcal{C}\ell_K\to\mathcal{C}\ell_\k)$ of the $\ell$-group of logarithmic classes of $K$ capitulates in the compositum $FK$.

We compute the 2-rank of the wild kernel WK2(F) of a number field by constructing a 2-class group ad hoc. The main result generalizes in the more intricate case l =2 the canonical isomorphism l μ F ⊗ Z Cl F ≃ l WK 2 (F) established for odd primes l under the assumption μ l ⊂F in a previous article (cf. (Acta. Arith. 67 (1994) 335; Math. Z. 238 (2001) 335)). It involves a criterium of triviality for the 2-part of the wild kernel of Galois number fields and, in the particular case of quadratic fields, it leads to a logarithmic interpretation of the diophantine conditions obtained by other authors.