Jax can be played in many different ways all forming around his ability to deal massive damage to a single target while also keeping the whole teams dps regulated with his stun.

The build I created makes Jax a forimdable hybrid and more importantly grants him supreme survivability.

So in this build I will be primarily focusing on Jax's early game nuking and late game survival in team fights. The reason I go for many items that dont give him a huge boost in damage is to keep him alive regardless of his passive.

Jx's passive is great because it gives you hp for getting items that boost your damage and end game you should get at least 600 extra hp from just his passive. But it is not enough to make him a champion that can be a glutton for punishment but also being a strong foe.

Pros
-Late game jax can easily 1v3 a team in TT especially with the right team comp
-Jax can harass and chase well
-Farming mid game is a breeze as well as backdooring
-Jax can kill tanks since he does 2 forms of damage effeciently
-Squishies lose their distance from Jax quickly making them an easy kill
-In team fights Jax becomes a formidable force once he acquires Guinsoos

Cons
- Early game farming is going to take time to get used to
- Early game kills can only happen with a good partner
- Early game harrass damage is strong but low mana and no escapabilty can mean an early death
- Jax is prone to being focused in team fights
- Chasing early game is hard to do without CD reduction
- Stun is useful but hard to acquire unless you are being attacked by melee dps
- JAX CANNOT INITIATE NO MATTER WHAT
- Jax is squishy until acquired both lifesteal and spell vamp

Like said above Jax can do damage with every build you use but this build maximizes his damagage while also not sacraficing survivability.

Ignite and Exhaust are the two best summoner spells for Jax in this build though ghost, heal, and flash have their uses and match certain playstyles.

Ignite is a great tool early game since it can pick up kills youd have to leap strike and dive for while exhaust lets you take on multiple enemies and or chase down a runner with ease.

My masteries evolved around being a hybrid going for magic pen CD reduction Dodge armor pen attack speed and ability power per level.

The armor pen and extra dodge are covered in the rune page.
However the magic resist runes and attack damage runes have their use as well.
With the attack damage runes Jax becomes a 1 move nuke early game with his leap strike and empower hitting for at least 250 at level 5. while also being able to get 1 extra hit in due to the slight boost in attack speed. The Magic Resist is simply to help Jax survive and increase his ults usefullness since that also boosts magic resist.

Be cautious of your surroundings and dont solo unless you have to....
Jax needs minion farm and champ kills in order to succeed since this build is quite expensive compared to others.
Early game farming requires using empower to help last hit as well as just auto attacking. It would be best to play with a partner who has a strong stun such as sion taric or even other forms of CC like Singed.
The best laning partner ive been with in my experience was Morgana due to her bind and magic resist lowering pool. The endless amount of burst dps we had together was phenomonal and if I encounter a better combo I will make sure to add it/
Try your best to not dive headfirst into a battle early game because jax is extremely squishy until acquiring spell vamp and lifesteal.

1) Do not initiate even after you get Warmongs because Jax is focused down and this build is weak to CC

2) Focus down the squishies and make sure to watch your Counter Strike CD since it is the MOST USEFULL MOVE IN A TEAM FIGHT

3) Pop your ult regardless at the start so you can take any starting abilities and gain the hp back through spell vamp and lifesteal

4) Always stay on a single enemy preferably one your team is focusing even if you are under pressure from someone attacking you because they will just fuel your stun to spamming potential.

Mid game farming is easy for Jax just auto attack everything and get your passive ult and guinsoos up.

Mid game is where this built gains potential because you deal 2 forms of damage Ap and AD
Your dodge makes it easy to kill any melee dps hero and your ult gives you the ability to survive a nuke from champs like Brand and Annie

Dont be afraid to engage at half health on a squishy with full health. Leap Strike and Empower even the odds and if you have a stun stored then you have gained the advantage because you will get a stream of hp from all your abilities.

Make sure to get blue and red buff since they make you unstopable CD wise and damage over time.

Late game try to backdoor and free turrets because the farm helps and every minion kill counts.

Dorans Blade is the first item i pick and boy do I get some haters on it

Dorans Blade is the best starting item for jax because it gives him a great boost in damage survivability (its hp and its damage adds to jax's passive) and the little bit of lifesteal while laning goes a long way.

Malady as a last item?

Well Malady is a greatly under used item in LoL even though it does what black cleaver does but its cheaper ( it lowers MR not Armor )

This build gives you a lot of armor pen but little to no magic pen and lowering your targets magic resistance greatly help your ult since both work off of auto attacks
Both your ult guinsoos and malady help themselves.
Guinsoos boosts attack speed and AP both giving you a little bit of health from your passive and making malady form quicker.
Your ult makes you attack faster and boosts your MR as well as getting quicker stacks from Malady
Malady lowers MR so fast that by the time your ult hits it deal about 25% extra damage then it would have.....

Warmogs makes Jax unkillable because of all hp you additively get from your passive and the farming for it.

Atmas works with jax extremely well because of Jax's passive which gives health from overall damage and ability power while atmas gives damage based on health Its armor and crit also make it so Jax deals extremly quick and potent melee damage

Builds are nothing and cannot be proven to be good unless they are tested by someone with a different playstyle from my own. Please if you do try this build contribute to this page by posting how well you did, ways I can improve my build ( It is the first one ive made so constructive criticism wanted )
I will edit this build as time goes on and you all comment upon it because I want this build to be used as much as you want to trust it.