It is seldom the case where a champion is defined by his actions after joining the League of Legends rather than before. Such is the case with Jax, for whom the argument could be made that he is the most prolific tournament fighter currently at the Institute of War. Before joining the League, Jax was an unremarkable soldier-for-hire. For reasons known only to the former leader of the League, High Councilor Reginald Ashram, Jax was put on the top of the list of candidates to receive a League Judgment - the interview process that either accepts or rejects a prospective champion. His Judgment was the quickest in League history, where the Doors of Acceptance glowed and slowly swung open as soon as it began. Jax faced no recorded Observation or Reflection during his Judgment.

Jax proved himself to be an immediate terror in the Fields of Justice. The self-proclaimed ''Armsmaster of the League'' rattled off a streak of consecutive wins that to this day has not been matched. A number of summoners in the League grew concerned that the perceived objectivity of the League of Legends would be questioned by the presence of an unknown fighter who was unbeatable. For this reason, the new leader of the League (following Reginald Ashram's disappearance), High Councilor Heyward Relivash, created special restrictions for Jax to fight under. This was something the League had never done before, and something that has never been done since. The burly fighter responded by imposing his own special conditions; as a means of protest, he permitted himself to fight using only a brass lamppost. Neither the League's sanctions nor his own has affected his winning ways. The League has since rescinded its sanctions, but Jax has not; he fights and fights well with his trusty brass lamppost.

''Be advised - there has been an outbreak of lamppost-shaped bruises in the League of Legends.'' - Gragas

Jax may be a bit weak early game, but he completely dominates late game, becoming almost unstoppable, even without Gunblade.

Pros:
- Unstoppable 1v1, and can solo multiple enemies with some items
- Great asset to team
- Great sustain
- He is just plain BA

Cons:
- A bit weak Early Game, needs some farm which may be difficult to get
- Leaping strike is the only ranged ability he has to avoid kiting
- Expensive items

For runes, the dodge yellows are a must, the other stuff is a bit more preferable but I get mpen and arm pen since he has both AD and AP

I take 0/21/9 mainly for the Ardor skill which is nice with the dodge runes and his e

Hextech Revolver: excellent early game item, gives your abilities some damage and great sustain. Every third auto attack deals magic damage so you get spell vamp from that and your abilities

Bloodthirster: this item is great for AD and lifesteal, makes Jax really hard to kill

Ninja Tabi: synergizes with dodge runes, mastery and his e skill

Will of the Ancients: Great for every reason the revolver is good for, except it gives an aura which is even better in team fights.

RageBlade: With the Ultimate's passive, you can easily get to 2.3 attack speed plus this item gives a nice AD and AP boost

Atma's Impaler: Not only does this item increases Attack damage, it increases health because of Jax's passive. Also makes backdooring super fast.

Optional:
Quick Silver Sash: If the enemy team is CC heavy, this might be a great item to get early on

Force of Nature: if the enemy team is stacking AP get this

Warmog's: not really needed but it works great with Atma's Impaler

I take the first skill in leaping strike because it allows you to harass the enemy champs in your lane. Leaping strike actually does more damage at the first point then empower, so it is best to get it first. Get only 1 point in e and max it last because with dodge runes and masteries you don't need the passive, and you use it for the utility. Level up the Ultimate every chance you get.

I get exhaust and ghost on Jax because both are great for escaping or chasing down an enemy with low health.

Ignite: If you can't get the last hit on that champion, get this.

Teleport: if you feel you need the mobility, then why not.

Flash: you can get this if you like it better then ghost, matter of preference

Cleanse: if the enemy has Heavy CC you might wanna pick this up

Smite: you aren't jungling

Heal: you dont need it

Clairvoyance: best left for support

Clarity: I don't think Jax has mana problems

Fortify: I don't think it lasts long enough to be all that useful

Rally: I never saw this ability be used effectively so I wouldn't recommend it

Revive: This build is not all you need if you die so many times that this is useful

Early Game:
With ability tome and leaping strike, you should be able to harass the enemy from farming as well. Also try to farm as much as you can. When you need to back pick up another ability tome or revolver if you can afford it. At this point you should be around level 6 or so, with the extra AP and sustain, you should be able to start forcing the enemy back if not getting kills on them.

Mid Game:
By this point, you should have Ninja tabi, and bloodthirster. If you have a few kills on you, start Roaming, clearing the enemy jungle, and your own if you have no jungler, and getting kills by ganking.

Late Game:
At this point you should have up to RageBlade, and maybe up to Atma's impaler. You are unstoppable, no single enemy can stop you, and you can even solo several if you have too. If their team is too great for yours, just backdoor like a boss

This build is intended to offer a decent balance between sustainibility and damage output. If you have any constructive criticism, You are welcome to comment.