  The behaviour of the quark coefficient function for the longitudinal
structure function F_L in deep-inelastic scattering is investigated for large
values of the Bjorken variable x. We combine a highly plausible conjecture on
the large-x limit of the physical evolution kernel for this quantity with our
explicit three-loop results to derive the coefficients of the three leading
large-x logarithms, alpha_s^n ln^(2n-1-k) (1-x), k = 1, 2, 3, to all orders in
the strong coupling constant alpha_s. Corresponding results are derived for the
non-C_F part of the gluon coefficient function suppressed by a factor 1-x, and
for the analogous subleading (1-x) ln^k (1-x) contributions in the quark case.
Our results appear to indicate an obstacle for an exponentiation with a higher
logarithmic accuracy.
