  Let $\ell$ be a rational prime number. Assuming the Gross-Kuz'min conjecture
along a $\Zl$-extension $K\_{\infty}$ of a number field $K$, we show that there
exist integers $\mut$, $\lat$ and $\widetilde{\nu}$ such that the exponent
$\tilde{e}\_{n}$ of the order $\ell^{\tilde{e}\_{n}}$ of the logarithmic class
group $\Clog{n}$ for the $n$-th layer $K\_{n}$ of $K\_{\infty}$ is given by
$\tilde{e}\_{n}=\widetilde{\mu}\ell^{n}+\widetilde{\lambda} n +
\widetilde{\nu}$, for $n$ big enough. We show some relations between the
classical invariants $\mu$ and $\lambda$, and their logarithmic counterparts
$\mut$ and $\lat$ for some class of $\Zl$-extensions. Additionally, we provide
numerical examples for the cyclotomic and the non-cyclotomic case.
