  We present all-order predictions for Higgs boson production plus at least one
jet which are accurate to leading logarithm in $\hat s/|p_\perp|^2$. Our
calculation includes full top and bottom quark mass dependence at all orders in
the logarithmic part, and to highest available order in the tree-level
matching. The calculation is implemented in the framework of High Energy Jets
(HEJ). This is the first cross section calculated with $\log(\hat s)$
resummation and matched to fixed order for a process requiring just one jet,
and our results also extend the region of resummation for processes with two
jets or more. This is possible because the resummation is performed explicitly
in phase space. We compare the results of our new calculation to LHC data and
to next-to-leading order predictions and find a numerically significant impact
of the logarithmic corrections in the shape of key distributions, which remains
after normalisation of the cross section.
