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 The League has since rescinded its sanctions, but Jax has not; he fights and fights well with his trusty brass lamppost.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

Greater Mark of Insight :

i think Jax needs some magic penetration early game.
magic penetration cause your hero doing full damage with his abilities Leap Strike and Empower and Counter Strike and Relentless Assault to your enemies.

Greater Seal of Evasion :

You need some dodge chances to activate Counter Strike Ability.
Counter Strike is a very important ability cause its an AOE stun.
It can fail your enemies channeling spells and gets your team an advantage to do more damage to your enemies also its an advantage for you to reaching higher attack speed cause your ultimate ability Relentless Assault.
Also dodge improves your magic resistant when you activate Relentless Assault too.

Greater Glyph of Potency :

Higher ability power causes you doing higher damage with Leap Strike and Empower and Counter Strike and Relentless Assault.
Also ability power can increases your health points cause your passive Relentless Assault ability.

Greater Quintessence of Potency :

I prefer ability power for quintessences too.
when you comparison Greater Quintessence of Ability Power with Greater Quintessence of Magic Penetration or Greater Quintessence of Evasion , you will find Greater Quintessence of Ability Power is more logical.
Cause it has 4.95 ability power stead 1.89 magic penetration or 1.5 dodge.

Summoner's Wrath :

Summoner's Wrath Improves Exhaust, Reduces target's armor and magic resist when you cast Exhaust on them.
Also grants you 5 ability power and 5 attack damage when Ignite is under cooldown.

Mental Force :

You must get 3/4 of this mastery.
Mental Force grants you 1/2/3/4 ability power.

Alacrity :

Alacrity grants you 1.5/3/4.5/6% attack speed.

Sorcery :

Sorcery grants you 1/2/3/4% cooldown reduce.

Deadliness :

Deadliness grants you 1/2/3/4% critical strike chance.

Weapon Expertise :

Weapon Expertise grants you 10% armor penetration.

Arcane Knowledge :

Arcane Knowledge grants you 10% magic penetration.

Havoc :

Havoc increses your damage dealt by 0.5/1/1.5%

Lethality :

Lethality grants you 10% critical strike damage.

Sunder :

Sunder grants you 2/4/6 armor penetration.

Archmage :

Archmage increses your ability power by 1.25/2.5/3.75/5%

Executioner :

Executioner increses your damage dealt by 6% to targets who below 40% health.

Doran's Shield :

I start early game with Doran's Shield it can help me survive early game in lane cause its health points regenerate per 5 seconds.
It's very important for Jax to start battles with full health points, DON'T FORGET IT.
also its armor and health points can help us when we struggle with minions cause Jax needs to struggle with minions to activating Counter Strike ability.

Health and Mana Potions :

Early game you need these potions cause you must start battles with full health and mana points, so try to keep your health and mana at maximum.
Every time you lose much Health points or mana points you must going back to base and recover health and mana, then buy some potions and going back to fight again.

Ninja Tabi :

Ninja Tabi is my favorite boot for Jax.
Its dodges helps you so much, cause it can activate your Counter Strike ability.
Counter Strike can reclaim you from a gang or shackle your enemies getting away.
Also it can helps you for destroying towers.
Yeah helps you for destroying towers. ;)
How? -Read Basic Play of this guide.
Also it increases your armor.

Sheen :

I like Sheen.
Early game Sheen is so important for you, cause it gets you more mana points and more ability powers.
Also its amplify your physical damages after using your abilities.

Rylai's Crystal Scepter :

It's your main item for ganging.
It slow your target's movement speed by 35% when using Leap Strike and Empower or after every 3rd attacks cause your Relentless Assault effect,
Or by 15% when using Counter Strike.
Also Rylai's Crystal Scepter gets you much health points and ability powers.
Now it's your time, you can steeve opponent champions when they are alone.
Always you must know where are the enemy champions, next you must know who is meat.
Then you can feed by them easily.

Lich Bane :

Convert Sheen to Lich Bane now.
Lich Bane gets you more ability powers,
Also it grants to you some magic resistant and movement speed.

Hextech Gunblade :

Now you are so powerful, so you must survive more.
Hextech Gunblade grants you spell vamp and life steel.
Also it gets to you some attack damages and ability powers.
At last you can activate it to doing 300 magical damages and slowing your target champion by 50%.
Don't forget to activate it, it's a very good advantage.

Rabadon's Deathcap :

It can Converts you to a super killer, or maybe 1 shot killer for mages ;)
Rabadon's Deathcap boosts your ability power so much.

Void Staff :

Void Staff Grants you 40% magic penetration and 70 ability power.
So you can do full damage to your enemies and be a fast killer.

Guinsoo's Rageblade :

You can try Guinsoo's Rageblade too.
It gets you some attack damages and ability powers.
Also it increases your attack speed by auto attacks or spell casting.
This effect is so powerful when it combines with Relentless Assault effect.
Relentless Assault increases your attack speed too also every 3rd attacks doing much magic damages to your enemies.

Thornmail :

You need Thornmail when your opponents have op ad champs and u must forget Void Staff

Banshee's Veil and Guardian Angel :

Banshee's Veil and Guardian Angel are very good items when your team didn't have any tank and you must play as tank.
These items gives you some defensive points.
but usually you didn't need them.

Equipment Mastery :

Relentless Assault is your passive ability.
Jax gets 300% of attack damages and 200% of ability powers of items as health point.
your health point increases by attack damage or ability power items.

Leap Strike :

Jax can jump to friend or enemy units in 600 range with Leap Strike ability.
If you jump on enemy targets you deal 100% of your attack damage and 70% of your ability power as magical damage to that target.
It's cooldown is reduces by every rank.

Empower :

When Empower is activated, your next attack deals 60/95/130/165/200 damage + 40% of your ability powers and 40% bonus attack damage, as magical damage.
Also Empower Resets your auto-attack.
It's cooldown is reduces by every rank.

Counter Strike :

Counter Strike increases your dodge chance by 10/12/14/16/18%.
Within 7 seconds after dodging an attack you can activate it for stuning all enemies in 400 range for 1 second and dealing 80/110/140/170/200 + 60% of your ability powers as magical damage to them.

Relentless Assault :

Relentless Assault is your ultimate ability.
Relentless Assault passively increases your attack speed by 6/10/14% for 2.5 seconds after your last auto-attack, it can stacks up to 10 times.
also every 3rd attacks deals 140/170/210 + 70% of your ability power as magical damage.
Activating Relentless Assault grants you 20/35/50 + your dodge percent, magic resistance for 5/6.5/8 seconds.

Exhaust :

Exhaust slows enemy target and reduce target attack damage.
If you get Summoner's Wrath it reduces target enemy armor and magic resistance too, so you can kill target easier with this summoner spell.

Ignite :

Ignite deals 50 true damage instantly and more 20 damage per level in 5 seconds too.
Also its reduces target's healing and regenerations over this time.

Cleanse :

Cleanse is good for getting rid of annoying blinds, stuns, exhausts etc.
It can save your life.

Flash :

Flash is Perfect for escaping through walls.
Flash is a good summoner spell for every Champion.

Leap-Empower :

This will be your main combo from level 2 onwards.
It allows you to jump on someone and deal a ton of damage with just one move.
It's very simple, just activate Empower and then use Leap Strike on your target, and bam you fly through the air to hit them with an empowered staff, and because empower resets your auto-attack timer, you get an extra auto-attack as well.
You can replay this combo every 5 seconds.

1. Early game you must focus on last hit of minions to collect more golds.

2. Using Leap-Empower combo for reducing enemies health under 40% then struggle with minions and when Counter Strike activated you can kill that enemy using Ignite and Exhaust and Jax abilities.
* Don't Forget you need 145 mana point for using Empower and Leap Strike and Counter Strike *

3. When lane is empty you can destroy towers :
Go back of tower and start attacking tower, when enemy minions come they begins attacking you, don't attack them and let them attacking you when you dodge using Counter Strike.
Using Counter Strike activates Sheen for more attack damage and beats the minions.
Also this way you prevents struggle between your minions and enemy minions and let your minions to focus attacking tower.
Empower didn't affects towers but over this time use Empower when its ready cause it activates Sheen effect again.

4. You must always knows where are the enemy champions :
Find vulnerable enemies and steeve them when they are alone.

5. Never initiate team fights :
Let tanks start the fight, then choose your target wisely.
You must 1st wreck champions that dealing much aoe damage to your team like Veigar and Ezreal.

6. Try to gang enemies who killed your teammates much with a teamwork.
It's so important to don't let them to be more powerful.

1. Never initiate team fights :
Tanks must initiate battles, You will die if you initiate a team fight.
You are an assassin, to fly out of bushes and the mists to 2 hit their casters, not someone to take the brunt of the damage.
The only time you should every initiate is perhaps for a gank, or if it's 1v1 and you know where the other 4 team members are.

2. Don't think you're invincible :
I have this problem, i thought i have an invincible champion but now i know i have only tenacious champion not invincible one.
Don't Forget if you do crazy things maybe all teammates die and nobody survive to defend and your team will be defeated easily.

3. Your Stun is not constantly :
Yes you have a stun but you can't stun constantly, and if they slow you chances are their team will jump on you and you will die.
be wise, struggle with mobs and when your stuns ready you can jump on low health enemies and stun all then kill that champion and come back fast.

You must be wise and know position of enemies every second for playing with Jax.
Some wards can help your team to win.
And don't initiate battles.
I believe Jax is a bug champion and he can change game result.
So don't try jax if you are noob.

Riot deletes all Greater Seal of Evasion.
So from now i use Greater Seal of Attack Speed.