  We numerically study the wetting (adsorption) transition of a polymer chain
on a disordered substrate in 1+1 dimension.Following the Poland-Scheraga model
of DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops.
This allows us to consider chain lengths of order $N \sim 10^5 $ to $10^6$,
with $10^4$ disorder realizations. Our study is based on the statistics of
loops between two contacts with the substrate, from which we define Binder-like
parameters: their crossings for various sizes $N$ allow a precise determination
of the critical temperature, and their finite size properties yields a
crossover exponent $\phi=1/(2-\alpha) \simeq 0.5$.We then analyse at
criticality the distribution of loop length $l$ in both regimes $l \sim O(N)$
and $1 \ll l \ll N$, as well as the finite-size properties of the contact
density and energy. Our conclusion is that the critical exponents for the
thermodynamics are the same as those of the pure case, except for strong
logarithmic corrections to scaling. The presence of these logarithmic
corrections in the thermodynamics is related to a disorder-dependent
logarithmic singularity that appears in the critical loop distribution in the
rescaled variable $\lambda=l/N$ as $\lambda \to 1$.
