Cracks (checks) in outside logs, particularly upward facing ones 1/2-inch and wider on walls without adequate overhangs to protect them from rain should be properly filled.
Install a tightly fitted piece of foam backing (available in thicknesses ranging from 1/4-inch up to several inches. Push the foam backing down approximately 3/8-inch below the surface of the log to allow the thickness of the filler.
It can then be troweled with a 3/4-inch-wide chinking knife to provide for a neat and proper sealing of the check.
A good wood cleaner can help remove watermarks, particularly on unstained wood, as long as they’re on top of the stain and not embedded in or under it on bare wood.
More important is to find out where the water is coming from and resolve the source so no long-term water infiltration leads to potentially damaging rot.