Hi,
Can't really help with the "log" part but my thicknesser - Ryobi - (13ins, I think) does say that the minimum thickness is 7mm. I'm guessing but I think that is because the 'tray' has upstands either side and would probably foul the cutters. What I read somewhere as an answer is to insert a board (laminex type, formica) and use that as the base. It does mean that any measuring scale on the machine is 'out' but I usually check the thickness anyway. Two battens - one each end - holds it in place. Depending on the wood I've got down to 1.5mm and a little under. As usual any alteration needs care but I've never had any problems other than some of the very thin stuff (again depending on the wood) shattering. I don't have a drumsander which probably wouldn't cause the shattering (which hasn't happened too often).
Hope it helps.
Bob H.
p.s. Forgot to mention that with no upstands to the base the wood can slide across - I think this is why the machine tray (base) has them - but it has never been a problem for me.