  The Welschinger numbers, a kind of a real analog of the Gromov-Witten numbers
which count the complex rational curves through a given generic collection of
points, bound from below the number of real rational curves for any real
generic collection of points. By the logarithmic equivalence of sequences we
mean the asymptotic equivalence of their logarithms. We prove such an
equivalence for the Welschinger and Gromov-Witten numbers of any toric Del
Pezzo surface with its tautological real structure, in particular, of the
projective plane, under the hypothesis that all, or almost all, chosen points
are real. We also study the positivity of Welschinger numbers and their
monotonicity with respect to the number of imaginary points.
