  We investigate first-order approximations to both i) Tsallis' entropy $S_q$
and ii) the $S_q$-MaxEnt solution (called q-exponential functions $e_q$). It is
shown that the functions arising from the procedure ii) are the MaxEnt
solutions to the entropy emerging from i). The present treatment is free of the
poles that, for classic quadratic Hamiltonians, appear in Tsallis' approach, as
demonstrated in [Europhysics Letters {\bf 104}, (2013), 60003]. Additionally,
we show that our treatment is compatible with extant date on the ozone layer.
