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Hello, I am UVATroller, and this is my second guide. My first was on support Chogath :) Constructive criticism is appreciated, and I Garentee you will like this (you will, right?)
The champion spotlight gives you great tips, the build is outdated however.

Jax Champion Spotlight

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Pros

+ Counter Strike negates monster damage and champion aa +can gank from behind walls like Fiddlesticks and can escape that way too, with crazy short cooldowns +has AOE stun +great clear speed +can duel almost anyone in the game when equally fed

Jax has an amazing kit, with a wall jump, aa reset, AOE stun that also negates AAs, and an amazing ult. He can duel almost anyone in the game, and uses a lampost for a weapon.

Pros

- No poke - burst damage from mages make this guy sad - hard CC makes this guy sad too - You have to buy a skin to get a real weapon :(

Jax has no poke, which isn't a problem when jungling. Since we are not building AP, Grandmaster-At-Arms doesn't give as much MR. Jax is really vulnerable when hard cc'd, and you have to shell out real money to give this guy a real weapon.

• Greater Quintessence of Attack Speed: Jax scales well with attack speed, with the synergy of his passive and ultimate.
  
• Greater Mark of Attack Damage DAMAGES
  
• Greater Glyph of Scaling Magic Resist: You don't need MR while jungling, and they will help you block some of that AP burst late game.
• Greater Glyph of Cooldown Reduction These help your Counter Strike to be off cooldown earlier.
• Greater Seal of Armor: Mandatory Seals for jungle clearing. Without them you won't be able to do even one full clear without backing.

21 Defence 9 into Offence

A T T A C K

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M A S T E R I E S

// AD Jax benefits more from the defense page than the offence page, while AP Jax benefits more from the offensive page.

Choices for Keystone Masteries

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The changed Trinity Force is amazing. It gives more CDR and more attack speed, which is exactly what Jax needs. All the stats Jax requires is in this item, the only time you won't get it is when you go AP or you decide on iceborn gaunlet

The new Guinsoo's Rageblade is a must buy if ahead, as it gives the old Sated Devourer passive.

Let's compare all the jungle items!

Stalker's Blade is the usual jungle item that you get in order to chase down and escape other people. It helps make up for the lack of CC Jax has.

skimerisher's saber skirmerisher's saber is the choice if you are facing 1 v 1 opponents constantly, and are split-pushing a lot. According to Op.gg, in Plat and above, skirmerisher's saber has a higher win rate, a 3% difference, so perhaps it is better that way.

Enchantment: Cinderhulk Jax is not a good tank, and scales heavily on attack damage and ability power. There is no real reason for you to pick Jax as a tank for your team, but if you really want you can build Titanic Hydra, Sterak's Gage, and Dead Man's Plate, if you want to go the off-tank bruiser build. Just remember that the build is weaker than the Ad build, and weaker than other Ad bruiser/tank junglers like Jarvan IV.

Enchantment: Bloodrazor The new enchantment for attack speed junglers, Bloodrazor gives tank destroying early game power in exchange for losing the double on-hit procs. It helps boost the power of Grandmaster-At-Arms and a lot of on-hit effects. You want to get this on Jax pretty much every game. The whole power of Jax's kit is in the attack speed synergy, which Bloodrazor provides.

Enchantment: Runic Echoes Riot has again changed a jungle enchantment. Ap jungle Jax is now unviable because of the removal of Runeglaive. If you are going for a cheese build with [luden's echo] and [rabadon's deathcap], you might as well play Ap Jax mid or top.

The most boring enchant of all, Warrior. Rito will no doubt rework this item to make it more interesting, but for now it's just a flat stat item. Warrior is better is you do not have any attack speed synergy items, or if you are a high Ad-scaling caster, but Enchantment: Warrior is worse on Jax.

Defensive Items

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Jax is very strong with Sterak's Gage, as he is a bruiser and diver. The health is really good if you are facing a mixed-damage team and you need one defensive item.

Iceborn Gauntlet partially fixes the problem with Jax having no lockdown. It works well with Empower, as it allows you to rapidly spam AOE slows. However, you cannot use this with Trinity Force, which really reduces the opportunities to buy this item. Does not stack with Enchantment: RuneglaiveHahahah Rito you change items so quickly now I have to edit this again

I get this when the enemy team is stacking armor for the up to 25% armor shred it can offer. Armor shred is awesome because it also increases the damage your teammates deal as well. The AD isn't bad either. However, similar to Iceborn Gauntlet, this item does not stack with Trinity Force, which reduces the opportunities to buy this item, usually with Iceborn Gauntlet

This is the go to item for double AP teams, and provides needed MR, and the passive is great combined with life steal from Blade of the Ruined King.

Boots

Heavy CC or most AP enemy team. Buy this if no other options are good.

Fed enemy ADC or most AD enemy team and you lost early game.

Lots of roaming or split pushing/ganking

Ap Jax

I get this item for Ap Jax because it has perfect synergy with this build, giving on-hit scaling magic damage, CDR, AP, and AS. Without this, it is difficult for Ap Jax to work.

This is what makes AP Jax's combo hurt even more, with Empower plus Grandmaster-At-Arms plus Lich Bane, you will easily one or two shot squishies. Now redundant because of Enchantment: Runeglaive Its funny how the line about Lich Bane being redundant is now redundent


Every game is different, one big mistake among low elo is people always build the same thing game after game, without changing the build to fit different situations

Jungle path

I'm not an expert on different jungle paths, so just watch the videos of these great youtubers (Totally not a paid promotion)
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40% CDR Jax jungle is an extremely mobile fast clearing jungler, He is less tanky than the other Jax builds, however, he makes up for it with his high mobility and the ability to negate a lot of damage from neutral jungle camps with his Counter Strike. I hope you like this guide, and don't forget to upvote it! :)

Edit history

11/15/2014 Guide published :)
11/24/2014 1k views thank you!
12/20/2014 Buffs to Jax! Lower cooldown on Counter Strike!
12/30/2014 25k views!
6/24/2015 Buffs to Jax! more stats!
6/25/2015 Revamp of guide
8/27/2015 Small changes for juggernaut items
11/24/2015 Added clarity on build choices and item choices.
9/5/2016 Fixed guide up a bit
11/21/2016 Remade guide for S7