A stack of round logs in a wall say 8 ft high will shrink more than an 8 ft log standing on end next to that wall. Put another way, a round block 12" in diameter and 12" long will shrink more in diameter than it will in length. How much a log will shrink depends on how dense the wood is and how much moisture is in the wood at the start, so there is no formula I know of to estimate that shrinkage.
Personally I would not worry to much about it on a small cabin. If, after things get done drying and settling, the center pole is too long, cut a half inch or so off the bottom end of it and let her settle, or if it's only a slight amount, live with it.
One thing you will learn with logs, they are always moving. With heat in a cabin and high humidity outside, a log will bend slightly from end to end. When the outside wood absorbs moisture it will swell, while the dry wood on the inside shrinks. The longer the log the more noticeable this becomes. Putting wood sealer on the logs after they are seasoned well will reduce this effect to some extent, but never completely overcome it. Letting snow build up against exterior log walls will soak the wood outside and can cause a huge amount of movement. I've seen this situation bend a 24 foot log as much as 2 inches from end to end. Such excessive movement can wreak havoc on an interior framed wall or cause doors to jam to the point where they will not open. You need to have eves long enough to keep snow from ever touching the exterior walls.
But, get a small cabin dried out good, and put sealer on it, and you won't have any noticeable problems.