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This will get me about 33% critical damage and 7% critical chance.
You can use dodge/damage runes, which would probably do even better, but as I am poor, I couldn't allow myself some.
Leap Strike and Relentless Assault have an 1:1 AP scale ratio, Counter Strike has about 0.8:1.
This is the build you should always follow - Relentless Assault > Counter Strike > Leap Strike > Empower.
The 25 damage increase and cooldown reduction on LS will be negligible, as you probably haven't got enough mana to spam it anyways. But 2% dodge increase and some damage on CS will help with survivability.
Only exception to the above is first three levels, where you should get
1. Leap Strike
3. Counter Strike
This is for good early-game harassment.
These are the items I've been using for a while, and done good in games. There are many-many ways to play Jax, but this is my way.
And if the game is dragging on, you should get Infinity's Edge (4080g). Against tank heavy teams, switch Malady with Madred's Bloodrazor (~3800g) (still start with Malady, but late game sell it and get Madred's).
The final build up should look something like that:
So, why I'd choose any of these items.
Malady - Second best attack speed item in the game, and lifesteal and passive makes it even better for the money.
Ninja Tabi - The only problem I've encountered with Jax is survivability ( duh) and some extra dodge and armor would only do good.
Frozen Mallet - 700 Health, 23 damage ( with the passive) and slow. Great survivability, fair damage increase and increased chasing ability... What's not to like?
Guinsoo's Rageblade - This is the item that I like, and it has a great synergy with Jax's ult. Both increase in power the more attacks are made, easy as that, and it also gives good stats. 35 ( ~40 damage), 45 AP and 4% attack speed and 6 AP per attack, stacking at 8 stacks ( 32% attack speed, 48 AP ). Thanks to this item you'll be going from 1.3 attack speed to 2.5 in a matter of seconds, not to mention the 1:1 AP scaling with ultimate.
Nashor's Tooth - 30% attack speed, 55 AP, 12/5 mana and -25% cooldowns. A great overall item.
Infinity's Edge - 80(92) damage, 20% crit and more critical damage. With such runes as mine, this really does great.
Possible items - Phantom Dancer, Banshee's Veil, any health/magic resist item.
I'm not going on about the game phases, how you should do something etc. Just don't feed, get the dragon at about level 6/7+ as often as you can and you should be fine. Dieing ain't bad, as long as it benefits the team.
Again, he can be played differently, but this is the most fun/efficient build I've tried.
I'm sorry if it has gotten too long already, but any comments or feedback is greatly appreciated.