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You always want to lane against ad champions, as jax eats them very easily.
If ashe//ezreal//trist//twitch//etc are mid, you dib it, swap, do whatever to get that lane.
Harass with leap from level one, depending on if they run or not you will get either empower (if they run) or counter strike (if they don't) at level 2, the one not chosen will be level 3.
Side lanes, people with snares//stuns are your best friends, just make sure you never lane against casters... Most of them, such as cassiopeia, karthus, anivia, will somehow always manage to eat you... (go figure)
If you're against one ad and one caster, ignore the caster, even if you get a kill and die it's worth it, you getting your items takes priority of stopping them from getting their items! (Unless you're middle, this rule only applies to side lanes, as they usually don't carry.)
These are the actual runes I use, they can get better, but this rune set does well for twitch, gang plank, trynd, jax, olaf, ashe, really any ad carry... (A infinity edge on any of them really compliments the runes)
Teleport and ghost are simply my favorites for jax, teleport I always use in 5v5, as it lets me go back to shop without losing out on much experiance, or to heal up and suprize another lane with a gank.
Ghost is simple, to chase down and kill everyone on their team... Cocky yes, wrong no.
Exhaust is a single target, good for one near sure kill, but ghost can be used as a better offence or defense summoner spell.
What skills to get when:
Lv 1: leap
Lv 2: (if they run when u leap get empower, if not get counter strike)
Lv 3: Opposite of lv 2
Lv 4: Opposite of lv 3
Lv 5: same as lv 3
Lv 6: Ult
Lv 7: Same as lv 4
Lv 8: same as lv 5
And repeat that pattern until empower and counter strike are maxed, along with your ult.
I usually chose to max leap last, as I like to save it to use as a tactical move in group fights, not for the damage output. As the normal lv 1 damage output combined with a powerful empower is good enough during lane phase.
Rally is not for jax, neither is fortify, exhaust, ignite, flash, clarity, or clarivoyance.
Rally, fort, clarity, and clarvoy shoudl be self explanatorily. o.o
Flash can be useful, but it's a safer bet to go with ghost.
Flash can only be used to dive a couple steps into tower, small amount compared to ghost.
Ignite is a single target spell, long cooldown and low damage compared to jaxs normal dmg... No real point since you're ganna eat everything anyways.
Exhaust once again single target is bad... for jax... they won't live long anyways... besides, if they always miss, you can't dodge.
Item order (as The item builder is weird)
Start with dorans shield and a health potion, any doran item can work depending on the situation, but you can't scope out your opponents style immediatly, so go with the shield, it's the safe bet.
When you go back you want to have atleast enough for ninja tabi (don't go back with less).
Recall, quick buy, and teleport back to your lane. (If you have change, GET WARDS.)
By the time this happens you should be getting to level 6, you're ganna want to try and convince your team to put off the laning phase awhile, until about level 10, when you can simply eat everyone. In the general area of level 10, assuming you have been having a decent time last hitting minions, you should be able to recall for blasting wand and pick axe, the ingrediants for rageblade. If you have any extra change, get wards.
The rest is bought 1(or 2) ingrediant(s) at a time, depending on how the team fights go.
The next item of course should be gunblade, as the 20% 20% lifesteal spell vamp does wonders with your ult and rageblade. While you're building that, throw in a sheen for your empower/leap combo, using empower will add shens base dmg increase effect onto your leap. Not a game breaker but good for it's cost, and that's not all a sheen does of course.
Because It's rare to see a team without a caster since the sunfire cape changes, you will need that guardian angel, though a hex drinker is nice, it just won't cut it again most ap champs. (Karthus lay waste is over 300 so that shield would be gone in less then a second, anivia ult, heimer turret//rocket, nunu ice blast, even soraka or taric would be taking that shield down)
You now deal insane damage, and have great survivability thanks to the guardian angel, time to up your damage with crits, that's right time for a infinity edge, it'll take a good 5 to 10 minutes to get but man will it rock them!
So it's late game, you're about done your items and are saying goodbye to your sheen, they haven't quit yet, even though they have a couple turrets left, and you have all but your furthest layer...
This is where your personal opinion really comes in, there are alot of items you can get, phantom dancer really compliments the crit runes and IE but, frozen mallet adds needed health and slows them. Of course sword of the divine also works well with your ult//as/ap/spell vamp build, not to mention the no dodge for 8 sec function can hurt alot of the enemies tanks if they have tabi aswell. At this time, if you REALLY need health, ignore common practices, get a warmogs. Even if it's casters tearing you down, they will just get magic pen if you get a fon, warmog is the way to go. If you keep getting money you can even Just trade in all your items, get a tabi, four warmogs, and a atma spear. (It sounds stupid but when if the game lasts long enough so that you have that much money, it will allow you to tank them with insane hp to kill the buildings.)
This isn't the most detailed build, but it might help someone.
Remember, MAP AWARENESS IS IMPORTANT. (always spend ur change on wards when u go back, even if you have 500 gold, never leave excess gold in your inventory, yes this makes saving for the b.f sword annoying, but it could save your team from ganks, and even get you a quick jungle kill and buff)
Also, IF YOU WANT TO PICK JAX, MAKE SURE YOU HAVE A GOOD TEAM FOR HIM.
Bad team to pick jax in:
A team with no tanks.
A team with one tank but no support.
A team with two or more ad champions.
A team with all ranged champions.
A team with quick pickers.
Those are just some examples, team composition is just as important as map awareness, player skill is actually less important then the two factors previously mentioned.
(Though map awareness is part of a individuals skill, and team composition comes from the level of communication and will to comprimize between individuals)