  We simulate directed site percolation on two lattices with 4 spatial and 1
time-like dimensions (simple and body-centered hypercubic in space) with the
standard single cluster spreading scheme. For efficiency, the code uses the
same ingredients (hashing, histogram re-weighing, and improved estimators) as
described in Phys. Rev. {\bf E 67}, 036101 (2003). Apart from providing the
most precise estimates for $p_c$ on these lattices, we provide a detailed
comparison with the logarithmic corrections calculated by Janssen and Stenull
[Phys. Rev. {\bf E 69}, 016125 (2004)]. Fits with the leading logarithmic terms
alone would give estimates of the powers of these logarithms which are too big
by typically 50%. When the next-to-leading terms are included, each of the
measured quantities (the average number of sites wetted at time $t$, their
average distance from the seed, and the probability of cluster survival) can be
fitted nearly perfectly. But these fits would not be mutually consistent. With
a consistent set of fit parameters, one obtains still much improvement over the
leading log - approximation. In particular we show that there is one
combination of these three observables which seems completely free of
logarithmic terms.
