We introduce a notion of F-isocrystals with logarithmic decay and give a conjecture relating this notion to slope filtrations. When the unit-root subcrystal has rank one we prove this conjecture. Combining this with a monodromy theorem we give a new proof of the Drinfeld-Kedlaya theorem. We also prove a generalized version of Wan's conjecture on genus stability for towers of curves coming from geometry.