On $\ell$-adic logarithms of Gauss sums

For the Gauss sum $S(\chi)$ corresponding to a character $\chi$ of order $\ell^md$ of the multiplicative group of a finite field $\mathbb F_q$ of characteristic $p$, we obtain an approximate formula for the $\ell$-adic logarithm of $S(\chi)$. We construct a special basis in the group of logarithms and define modulo $\ell^m$ the coefficients of $\log_\ell(S(\chi))$ relative to this basis (modulo $\ell^{m-1}$ if $\ell=2$). These coefficients are defined in terms of power residues of some cyclotomic numbers at the places over $p$.