  A NNLO analysis of certain logarithmic expansions, developed for precision
studies of the evolution of the QCD parton distributions (pdf) at the Large
Hadron Collider, is presented. We elaborate on their relations to all the
solutions of the DGLAP equations that have been hitherto obtained from Mellin
space, to which are equivalent. Exact expansions, equivalent to exact solutions
of the equations, are constructed in the non-singlet sector. The algorithmic
features of our approach are also emphasized, since this method allows to
obtain numerical solutions of the evolution equations with the same accuracy of
other methods, based on Mellin space, and of brute force methods, which solve
the equations by finite differences. The implementation of our analysis allows
to compare with existing benchmarks for the evolution of the pdf's, useful for
applications at the LHC, and to extend them significantly in a systematic
fashion, especially when solutions that retain logarithmic corrections only of
a certain accuracy are searched for.
