Probably largely up to your taste and the palette you are comfortable with.
When you build with logs, there are two basic strategies. One is rough finished, the other is close finished.
With rough finish, you leave the logs generally round, only notching the ends and for openings. The only work along the length of the log is to debark it (bark is not a good finish, interior or exterior--it peels off in the weather).
The other way is close finished, which is seen usually only where the trees are quite huge. The classic example of this are Scandanavian log homes, which are often not but three "logs" tall (the logs being more than 1m in diameter). So, slabs are taken off the felled logs which are turned into other building parts, leaving a 30-50cm thick center.
Close fininshed logs will show their interior look, and will weather to similar colors as milled lumber--pale grays and hints of silver, over a tan sort of base color.
Rough finished logs retain their outer sapwoor, which has more of an umber base with highlights of red and amber.
In either case, the meeting surfaces of the logs, one to another, has a circular shape which leaves a gap. That gap is filled with "chinking" (also known as "daubing"), which is a mud and straw and clay mix (modern mixes use portland cement and flexible epoxies)
The chinking will have a pale gray clay color or it might be a mud-dark color (it will usually match the local "dirt" color to some etent). A touch of moss/mildew will crust the top curve of the logs, and up on the bottom edge of the chinking. Black-green is a good base for this, with additional greens to get to a comfortable mossy look.
For thatch roofing, I'd probably start with a base of khaki-drab, or whatever tan I had that had a greenish hue. I'd build up lighter tan colors over that, and finish with a drybroush of Radome Tan or Unbleached Linen.