  Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$
elements and let $\theta$ be a generator of the cyclic group $\mathbb F_q^*$.
For each $a\in \mathbb F_q^*$, let $\log_{\theta} a$ be the unique integer
$i\in \{1, \ldots, q-1\}$ such that $a=\theta^i$. Given polynomials $P_1,
\ldots, P_k\in \mathbb F_q[x]$ and divisors $1<d_1, \ldots, d_k$ of $q-1$, we
discuss the distribution of the functions $$F_{i}:y\mapsto
\log_{\theta}P_i(y)\pmod {d_i}, $$ over the set $\mathbb F_q\setminus
\cup_{i=1}^k\{y\in \mathbb F_q\,|\, P_i(y)=0\}$. Our main result entails that,
under a natural multiplicative condition on the pairs $(d_i, P_i)$, the
functions $F_i$ are asymptotically independent. We also provide some
applications that, in particular, relates to past work.
