  Let $\ell$ be a rational prime number and $K$ a number field. We prove that
the logarithmic module $X_{d}$ attached to a $\mathbb{Z}_{\ell}^{d}$-extension
$K_{d}$ of $K$ is a noetherian $\Lambda_{d}$-module. Moreover, under the
Gross-Kuz'min conjecture we prove that it is also torsion. We exploit this fact
to deduce local and global information of the logarithmic invariants
$\tilde{\mu}$ and $\tilde{\lambda}$ of $\mathbb{Z}_{\ell}$-extensions.
