So the guy has a rageblade and lichbane, two pure damage items.. Maybe try killing him? His ulti isnt on forever and only gives him so much defense.
He has no sustain, try poking and siegeing?
Your intial post was pretty ridiculous, I didn't even notice. You basically said "He blows his main chasing spell and his main defensive spell which has a long cooldown putting him at risk and he gets a favorable trade. Man, that's unfair." That's honestly flat out ridiculous.
From what it seems, you dont realize what tools are available to fight him and think that if you can't straight up brawl with him he must be a problem. The point of this being a strategy game and not CoD is that there's more than just buy damage and defense and attacking the enemy.
Like I said, Jax is item and level dependent. early game his only defense is on a long cooldown and easy to bait out or avoid. He has no natural sustain. He also has to be in melee range to actually do anything. Displacements are incredibly powerful against him, as are hard CCs. Since his stacks only last 2.5 seconds, pushing him out of a fight or locking him up repeatedly reduces a lot of his damage. Since he relies on repeatedly procing his ult to wear down targets, that disruption reduces the number of times he procs.
Since he is item dependent, there's a decent period of the game where he has to farm and can't afford to influence fights. Forcing action elsewhere on the map will result in either you getting an advantage because the jax keeps farming or the jax getting behind in farm because he goes to fight. again, strategy game.
Since he is full melee and isn't exceptionally mobile, he has to facecheck a teamfight to effect it. Using disengage properly can bait his abilities and make him take free damage. Positioning such that he is forced to DPS your meaty targets while your DPS hits him directly puts him in a difficult situation.
Instead of being angry because a champion designed to scale and be strong 1v1 can kill you, think of strategies you can use to prevent him from having those ideal situations.