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 The League has since rescinded its sanctions, but Jax has not; he fights and fights well with his trusty brass lamppost.

Hello, welcome to my Jax guide. I'm actually terrible at writing guides, so yeah. I guess I'll introduce myself. I am currently known as The Skilled Noob in League of Legends. Jax is my most played champion in ranked. I started playing LoL when it was in beta. I was trained in the arts of LoL by this one dude who was epic. He taught me, basically, how to own people. I gave my account away after LoL came out of beta and made a new account because I needed a new name. At first I was a scrub Jax, buying Ninja Tabi every game, building lots of AP, not knowing why I lost.... Until I met a friend who taught me how to itemize and, well, how to use strategy. I took this knowledge and forged my current state Jax, slowly though. I realized I was actually really good, and decided to embark on Ranked play again. I gave up before when I couldn't handle 1100 ELO. I carried myself most of the way, duo Queued some of the way. And to put the cherry on top, right before I got a bronze medal, I duo Queued with a 1600 elo player, and according to him, I did OK. It is my most memorable game to date, here is a picture of the last team fight:
http://i51.tinypic.com/2jayh49.jpg

I get critical strike chance marks and quintessences because:
Think of critical strike chance as a damage multiplier, the longer you stick to your target, the more critical strike chance does for you.
I get Magic Resist per level glyphs because they are just the best thing you could use in glyphs. Critical strike chance is too low. Critical strike damage peaks out at 17 extra critical hit damage. Because of the fact that you won't be getting Ninja Tabi most of the time, the dodge seals are important. Not for survivability, but because getting 1 more stun is better than 1 more critical hit. (changed from critical strike chance seals to dodge seals because I decided that the dodge seals were necessary)

I would just like to clarify, right here, why I choose to focus AD rather than AP. I will explain why exactly I trust critical strike chance to get the job done.

First of all, what is AP? Ability Power. Ability Power affects your Abilities. Common knowledge. AP scales with your abilities. Say you have 500 AP and your base spell damage is 200. 500 times 0.7 = 350. 350 + 200 = 550. Now let's say you have 300 attack damage, 265 percent critical strike damage, and 70 percent critical strike chance. You know you should be getting critical hits every 7 out of 10 attacks you make. Now, instead of scaling like AP, the critical strike chance MULTIPLIES the damage, and is also boosted even further by critical strike damage. 2.65 times 300 = 795. Not to mention the fact that you'll score critical hits more than you use your abilities, even your R. Still not convinced? Follow this link...
http://leaguecraft.com/builder/Jax/?items=70,108,89,60,64,108&runes=45,45,45,45,45,45,45,45,45,58,58,58,58,58,58,58,58,58,98,98,98,98,98,98,98,98,98,61,61,61&masteries=3100441130330103030000000000000000000000000&level=18

It is a standard anti-balanced team and anti-squishy team build. (see Items section)
Click on the Battle Theory tab.

912 DPS... Beat that with an AP focused non-glass-cannon Jax build.

I get mostly offense, and go a little bit into the defensive tree. Attack speed mastery is a must for Jax, because it can make the difference of an entire auto attack. Brute force can have a dramatic effect once you are getting critical hits like crazy. You must have the spell penetration mastery. Armor penetration is pure win. You need the things that go with your summoner spells. Getting +6 armor and magic resistance from 6 points into the defensive tree is really good!

I will spend a lot of time in this particular section, bear with me.

The Core Item Build

The Core Item build (the items that you will get every single game) are:

Hextech Gunblade
Phantom Dancer
Infinity Edge

These items never ever change. The other 3 items varies based on the enemy team composition.

First thing, you will either get Ninja Tabi or Mercury Treads. No other boots are allowed.

When Do I Get Mercury Treads??
A: You get Mercury Treads when the enemy team is balanced, or mostly/all AP.
When Do I Get Ninja Tabi??
A: You get Ninja Tabi when the enemy team is mostly/all AD.

For example, if there are 3 AD carries on the other team, you need to get Ninja Tabi.
If the enemy team has Garen, Olaf, Ashe, Nunu, and Zilean, you get Mercury Treads, because Olaf and Garen are not carries.

After you finish your first four items, there are 4 paths you can take.

1. Offensive
2. Defensive (ARMOR)
3. Defensive (MAGIC RESIST)
4. Defensive (BALANCED)

If you don't think you need any more survivability than what you have, and you're feeling like a bad-***, get Hextech Gunblade and Last Whisper.

If the enemy team is balanced and you want survivability, get Guardian Angel. You can then either get an offensive item (Gunblade or Whisper) or you can get another defensive item (Warmog's Armor, Frozen Mallet, or Rylai's Crystal Scepter).

If the enemy team is all AD, and you want survivability, definitely get Thornmail. If you want more survivability after Thornmail, you can get Sunfire Cape, Randuin's Omen, or Atma's Impaler. If you want to take the offensive path after Thornmail, you get Gunblade or Whisper.

Now if the enemy team is all AP, and you want survivability, get Banshee's Veil. If you want more survivability, which if they really are a bunch of casters I would think you will, you can get Force of Nature, Quick Silver Sash, or Abyssal Scepter. But if you want to stay on the offensive track after Banshee's Veil, get your Gunblade or Whisper.

I would recommend that, if the enemy team has, for example:
4 AP, 1 AD
4 AD, 1 AP

You get 1 defensive item, and 1 offensive item as your last two items.

Now if they are:
5 AP
5 AD
DEFINITELY! DEFINITELY! DEFINITELY! take the defensive path on both of your last two items.

In most cases, 3 AP and 2 AD is balanced, and vice versa.

If they have Gragas, Galio, Ashe, Twitch, and Master Yi... That is a AD focused team. So I would get Ninja Tabi, Thornmail, and Hextech Gunblade as my varied items.

Now if they are Gragas, Galio, Maoki, Ashe, and Tryndamere... That is a balanced team. So I would get Mercury Treads, Guardian Angel, and Last Whisper as my varied items.

Now if you noticed, I got a second Gunblade against the squishier team. And against the more tanky team, I got a Last Whisper.

If you are confused, this picture I made will help make what I have said more clear.

To me, this is a matter of preference. I typically use this order:

Q
W
Q
E

After that I upgrade focus on maxing out Q or E. Don't max W first ever.
Q gives you more harass in your lane.
E gives you more survivability.
You decide what you need.
Never skip out on upgrading R at level 6, 11, and 16.

Leap Strike

(Active): Jax leaps towards a target. If it's an enemy, he attacks it and deals magic damage.

This is what you use to harass in the laning phase. You will get harassed back, but that's what you bought 3 health potions for.

Empower

(Active): Jax charges his weapon with energy, causing his next attack to deal bonus magic damage.

You can use this as a combo with Q.
To do this here is what you do:
Press W. Press Q. Click Your Target. It will apply the damage of W and Q in one hit.
Also, Empower increases your attack speed dramatically for 1 auto attack. The cool thing about this ability is that you can use it to harass (W + Q) or sustain yourself (press W immediately after an auto attack).

Counter Strike

(Passive): Jax's prowess allows him to dodge enemy attacks frequently.

18 percent dodge is nice.

(Active): Jax stuns surrounding enemies for 1 second and deals magic damage to them. Usable only within 7 seconds of dodging an attack.

The fastest way to trigger your stun is enemy minions, USE THEM TO YOUR ADVANTAGE! Also, you can land a triple-harass combo. Get your stun activated, then activate W, Then use Q on your target, then use your stun. This way, the enemy can't even harass you back to, more frustrating for them, more fun for you! Remember that you have a time limit of 7 seconds to do this though.

Relentless Assault

(Passive): Jax's attacks buff his attack speed for a short duration, stacking up to 10 times. While having one or more stacks, every 3rd attack will deal bonus magic damage.

This stacks on towers but not on inhibitors, however, if you stack it off of minions and then start attacking an inhibitor, you keep the stacks! This passive is what makes Jax a true duelist.

(Active): Grants 20/35/50 + Jax's dodge percent in bonus magic resistance for a short time upon activation.

Built-in survivability against mages! COOL! :D

I use Exhaust and Ignite on Jax because his Q gives me enough of an escape.
However, some other good spells are: Ghost, Teleport, Flash, and Cleanse.
Whatever I haven't listed you should never take ever.

Exhaust: You can use it to stop somebody from finishing off your tower. It's actually real funny to use exhaust on someone who is one auto attack away from finishing your tower, because they will just keep hitting your tower even if you start beating them to death and you'll get the kill. That's why, no matter how close you are to finishing the tower, when the enemies come, you run! You can use it to stop somebody from killing you or your team mates! Or use it to stop somebody from running away.

Ignite: This is a good counter against champions like Swain and Mundo. It also deals a good bit of damage, and you can use it as a last hitter, or use it on somebody who just activated a health potion.

Ghost: This is a good spell for baiting people. Like when you're almost dead and you activate Ghost right as you reach your turret, the enemy is going to tower dive because you just blew his mind up. You can also use it to initiate a fight, there isn't anything like running into battle with all of your team mates!

Cleanse: Right now, this spell completely removes Exhaust, so don't immediately pass it up.

Flash: Awesome spell! Very fun if you ask me. Jump over pretty much anything. Use it right before the enemy strikes a killing blow! Or you could suddenly flash over to an enemy to strike a killing blow.

You can use Teleport to get gold and experience faster. And save towers. And backdoor! Teleport has underestimated utility if you ask me.

Pros:

A cool thing about Jax is that he does more damage than any other bruiser. As a matter of fact, he is the best duelist in the game. As soon as you're fed/farmed, the game will get very fun.

Cons:

Harassed easily.
Easy to burst down.
Doesn't bring a lot to his team until mid game.

Jax has good utility. His Q allows you to jump to your team mates. This means when Sion and Cho'gath gank your WW because he is alone, you can leap over the wall and turn the 2v1 into a 2v2! Also, when Twitch pops up at the other side of the wall to kill you and your team mates by shooting through the wall, you can leap over the wall and kill him before he can do a lot of damage! Jax, when built properly, becomes fairly tanky. You can help your tank initiate the fight!

This build is the best Jax build because it gets Jax's damage output to be very high while at the same time maintaining bruiser status. You won't be as tanky as Olaf though.

Like, late game, you will be SO tanky! If you get a GA and then a Warmog's Armor to be cute, you will be so tanky that the enemy team would be STUPID to focus you! Some people say Warmog's Armor is a bad item, but I disagree, and people that think WA is bad, are bad...

Did you know that Warmog's Armor, maxed out, is worth 10 kills? I bet not!

When you're playing Jax, you should be trying to get as much gold as you can. Jax is not a cheap champion to build XD.

Make sure you don't miss out on team fights! Politely ask your team mates to let you clear the huge enemy minion wave.

Don't spam your abilities on minions, except W. You want to last hit, not push.

When an enemy buys SoTD(Sword of The Divine), it may be best not to help your tank initiate a team fight. Because obviously if the enemy team has gotten SoTD, they really want you dead and will most likely focus you.

I don't know why there is a "Ranked Play" thing here... You shouldn't write a guide if you can't do well with it yourself at a good ELO in ranked games.

Once that you've obtained a Hextech Revolver AND Vampiric Scepter you can jungle without any HP potions. I wouldn't try to sneak the dragon, because the other team may have planted a ward.

Because the rageblade is a bad item. Sure, the statistics are cost-effective, but you are essentially gimping your mid-late game. To do a lot of damage, you have to get critical strike chance, and that is a fact. Also, you will end up selling the rageblade at the end anyways, don't waste your time.

Yes. Unless you want the build to just... not work lol XD.
Q: This is an AD focused build, so why not get armor penetration runes?
A: Because you need a lot of critical strike chance.

Well as I said before, if the enemy team isn't tanky enough for you buying armor penetration to be absolutely necessary, you should get another Hextech Gunblade. Also, a percentage of armor penetration is better because you aren't using armor penetration runes. So that is pretty much just strengthening the percentage because flat armor penetration is calculated BEFORE % armor penetration.
Let's say the enemy Dr. Mundo has 200 armor.
You want armor penetration? The Black Cleaver maxes it's reduction value at 45 armor penetration, leaving the enemy Mundo with 155 effective armor.
Say you got the Last Whisper, Dr. Mundo now has 120 effective armor.
Not to mention the Black Cleaver's attack speed statistic is not efficient on Jax at this point.

Yes, if you would like to lose, that is. AP Jax is pretty fail, because you don't have sustainability, only burst. And if the enemy team is any good, they will chain CC you if you even dare try to leap harass and then run away.

I hope my guide helps you to be a better Jax player. It's the build that matters anyways, and I can guarantee you it works. This is definitely the best build/guide I have made. I have made many failed attempts, ah-aha... Jax is my main nowadays after all...