Hello my name is sneaker12, I play on the N/A server and I love jax hes an amazing champion when played in any way whether its ap or ad or hybrid. My favorite way of building Jax is playing him AP.

pros

cons

Unyielding

Destruction

+ very high burst damage

- no health in the build

+ short cooldowns

- not as high sustain damage

+ a high sustain damage with ult

- weak early game/lane phase

+ very fun to play

- nemesis jax is no longer on sale

23/7/0
I get 23 in the offense tree because my build also uses attack speed in order to get more ultimate hits off every 3rd auto attack
In Defense:
I take 7 points in the defense tree because having health regen is good in lane and having health is good in the late game as u want to live for a little while longer in fights

In Offense:
I grab summoners wrath: for more ap and a bit more ad while ignite is on cd
attack speed cool down reduction and all the basic ap masteries

My runes focus on the early game to help jax in lane
Mark of Magic Penetration
I use marks of magic pen to cut through early game magic resist most top laners will have
Seal of Armor
most top laners are AD so having armor will help you to trade better early on
Glyph of Ability Power
Greater Quintessence of Ability Power
for ability power this will give you 34 ap at lvl 1 as long as you include my masteries, this will give you alot of early game damage which you need to get through the lane phase

Ignite
I use ignite because it will help to secure kills in lane and in fights when ignite is used if you have summoner's wrath it will give you 5 bonus AD and AP while the spell is on cd
Flash
flash is very useful as I tend not to carry wards with me so this helps me to flash to a champ to deal a finishing blow if Leap Strike is on cd or when i wish to get in range of a minion to use my leap
Ghost
if you do however carry wards with you ghost is a viable option i just personally like flash better, if you are good with ward jumps and always keep a ward in your emergency kit then open up that kit and put a ward on the other side of the wall to escape that pesky jungler that came in when you were in a 3v1 at blue

Leap Strike
this is Jax's catch up and get away ability while it scales very well don't use leap strike to often in lane cause it will burn through mana and also will leave u open for ganks

Empower
My favorite ability on AP Jax. When using empower use it as an auto attack reset when at all possible because this will apply 2 hits of your passive and leave you with only 1 hit away from an ult proc. most Jax's will use their leap strike and empower at the same time but you should use empower with an auto attack for more damage. Empower when combined with lich bane does the most damage in Jax's kit when added on with his grandmaster's might you may be able to 1 shot an adc but even if you your if you use your autos right in 1 more hit you should do upwards of 700-800 damage anyways

Counter Strike
This is Jax's engage tool you should use this whenever you're fighting anyone for both the dodge(which is super op) and for the stun there is no scaling on this so use this only for utility never max first or second

Master-At-Arms
Every 3rd auto attack someone dies is a pretty good idea of this ability, the AP scaling on Jax's ult is 70% of his AP plus 220 at level 16 this is an amazing burst of damage which is achieved easily because of empower. The activation of the ult will give Jax an immense amount of armor and magic resist but seeing as how magic resist scales on AP and armor scales on ad you will have more than enough MR reaching into the 200-300 bonus MR range with a full build so you can dive on the enemy AP easily you won't really miss the armor though because of your counter strike you'll be dodging all of the ad carries basics attacks, that is if he lasts more than 1.6 seconds.

I like to start Jax off with 2 different starts
1. Doran's Ring Health Potionx2
2. Cloth Armor Health Potionx5
the reason for these starts are simple if you are playing against someone who will likely poke and or kill you early on and who does ad damage then Cloth Armor is great for getting through lane phase. If you can win the lane without much trouble because they are not a good poke champion or if they are a AP champion then buy Doran's Ring this gives you a good boost in AP and health early along with some mana if you want to spam more abilities out for more harrass

Jax has 4 major spikes in damage for the most part its when you get your Sheen then your Lich Bane then 2 major spikes in your burst and sustain damage are Rabadon's Deathcap and Nashor's Tooth

Reasons for my items
1. Lich Bane the proc on lich bane when you use a w does an insane amount of damage similar to when Fizz uses Urchin Strike and a Lich Bane
2. Nashor's Tooth gives a great amount of attack speed along with AP and whenever you auto attack you deal damage based on your AP that adds alot to jax's sustained damage which lacks on auto attacks that don't Empower or Master-At-Arms
3. Rabadon's Deathcap rabadons gives an insane amount of AP so this will greatly increase your burst as you are playing jax as an assassin you will need to be able to 1 shot ad carries
4. Berserker's Greaves I use berserkers however you can use whichever lvl 2 boots you want all dependant on what you are missing or what your preferences are, when i play jax i rely on getting alot of empowered and ult procs off on my auto attacks so i want a high attack speed
5. Hextech Gunblade this will give you the spell vamp and life steal you needed this will be good for going into the jungle after a short skirmish healing up fast and coming back in to burst down another champion the active on hextech also has a nice slow and it scales on AP. when i use my w i usually get around 180 health back and every third hit i get another 180 back along with my spell vamp from my first auto and you can heal alot every combo
6. Guardian Angel this is the item no one ever wants to see on a jax ever, this will give you insane amounts of armor and magic resist to add to your insane amounts of armor and magic resist and it will also revive you seeing as how AP jax is all about burst you can very realistically come out of guardian angel with all your spells off cd and burst the enemy down before he even realized he took you down in the first place

in conclusion

these items give a great deal of ability power, alot of attack speed, some nice actives and on hit effects and a bit of tankiness. All this together will give you a tanky super bursty lamp wielding maniac with alot of sustain damage in between bursts

Lane Phase

To play lane phase as AP Jax you should focus on farming if the enemy comes near to you should abuse the fact that you have an auto attack reset and auto attack them then use your w for alot of damage. once you hit level 4 you should have all your abilities plus more damage with your w you should ask your jungler for a gank because if this happens it will most likely net a kill and secure you for the rest of the lane phase. once AP Jax gets ahead in lane he becomes a complete lane bully killing anyone who tries to fight him just make sure to bring a ward because they will most likely want to shut you down.

Team Fights

When team fighting you have 2 main roles dependant on your playstyle you can either be a initiator or you can be an assassin either works well but if you usually play an assassin then you should stick to doing that kind of a role with jax. when jumping into a fight your main role is to kill the enemies carries which becomes easy because of your Counter Strike because the enemy ad carry won't be able to kill you.
team fights can go a few ways dependant on a few things

1.

whos fed on the enemy team if its the ad carry you will most likely want to jump straight onto them. But you should wait until most of the cc is down in order to get all of your abilities off.

2.

how fed your are at this point if you absolutely crushed top lane then you can probably start a fight by yourself as long as your team is there for back up, just remember to use your jump on the ad carry or ap carry as your jump does alot of jax's damage.

AP Jax is just as viable as ad jax the only thing to keep in mind is who the player is if you feel you like to use your abilities and are more of a bursty Jax player then definitely give this a try you will probably like it. If you like to play ad Jax because of its high sustain damage and survivability then still give this build a try because its definitely fun and does do alot of damage.

This is my first guide and ill be updating it but i hope you like it please try out AP Jax because its a lot of fun and just as viable as ad Jax you just have to have an open mind and the ability to try a different playstyle on a champ.