I'm thinking Ajax hacks up Logan pretty good, but Logan should be able to last long enough to get in close to eventually kill Ajax.
Wolverine took a lot of punishment from Sabertooth with the bone claws, though he got trounced. It's really a question if Ajax hits enough critical areas (head, neck, midsection, hacking arms and legs) to disable Logan and then continuing the assault till he KO's/kills him.