Why Hanson's logarithmic scoring rule define an Automated Market Maker?

Why Hanson’s logarithmic scoring rule define an Automated Market Maker?
We have in this paper
in page 47, section automated market maker, that this scoring rule can "easily" be turned into an AMM. But it seems to be so easy that no one can explain that clearly. Sometimes, LMSR is directly defined as an AMM like here
middle of page 2.
Well, in all research I did, everyone consider that it’s an AMM, but Hanson do not really explain why…