Plumbing a log house can indeed be challenging. We've owned two of them, the second of which we built almost entirely by ourselves. We spent a lot of time in the design phase coming up with a layout that clustered most of the plumbing at one end and made maximum use of interior partition walls for vertical runs. Ours is two stories, with exposed beams on the first floor, so there were plenty of places for horizontal runs. In the rooms where that was done (bathrooms and laundry room), we put in tongue-and-groove pine ceilings to hide the piping.
One other thing we took into consideration was esthetics. We have a 12/12 standing seam metal roof, and didn't want to have to look at a bunch of vent pipes sticking up through it. Using horizontal runs, plus having a small attic space to work with, we were able to tie all the vents into the main stack, which is the only roof penetration.
We did use one AAV, for the kitchen sink. It's inside the sink cabinet. You cannot use AAVs in the crawl space. An AAV has to be higher that the flood level of the fixture being vented. And also note that AAVs are illegal in many places.