Well, im a relatively new player, and as it says in that sentence. NOT THAT EXPERIENCED
still, i thought to try out making my build, of my favorite player: Jax

Anyways, I'm absolutly open to the more experienced players, for input on how this build could work the best.
and i've been playing this way with jax for some time. one thing is he's hard to play early game, but as soon as i get the Rageblade, and the Gunblade, im good.

Oh yes, the Mastery tree and the Abilety sequence ive been inspired by BigDowny 's:
"Jax: Dodge, Dodge, Killing spree""

The sequence i think you should get items.

Cloth Armor

Total Price: 300
Recipe Price: 300

18 Armor.

Why NOT?:

Boots of Speed

Total Price: 350
Recipe Price: 350

UNIQUE Passive: Enhanced Movement 1

Because it doesn't do a thing, all you get to be is a little faster.

Next is of course:

Ninja Tabi

Total Price: 850
Recipe Price: 200

25 Armor, 12 Dodge. UNIQUE Passive: Enhanced Movement 2

Ninja Tabi is exstremely good for Jax, so far i know its the only item that boosts your dodge change by that, and, it have armor, and also, you move faster. Good for chasing other champions. since its cheaper than the others and you can get it fast.

And here we come for the item, designed for jax (I've been told)

Guinsoo's Rageblade

Total Price: 2235
Recipe Price: 400

35 Attack Damage, 45 Ability Power. Passive: On attack or spell cast, increases your Attack Speed by 4% and Ability Power by 6. Lasts 5 seconds (stacks up to 8 times).

Guinsoo's Rageblade is really the item you want, absolutly a must for jax. the recipe items you can decide for yourself what to buy first.
Blasting wand
Pickaxe


Next is:

Hextech Gunblade

Total Price: 3625
Recipe Price: 600

60 Attack Damage, 75 Ability Power, 20 Life Steal, 20 Spell Vamp. UNIQUE Active: Deals 300 magic damage and slows the target champion by 50% for 3 seconds (700 range). 60 second cooldown.

Now, you can't just buy this item, you will never get that much money and do good at the same time...well i wouldn't.
so thats why you need to get the recipes for it.
And what i think is the best choice to get is:
Vampiric Scepter
Then
Bilgewater Cutlassx2 Amplifying TomeAnd Hextech Revolver

And now you just get the rest of the items, now you can do pretty good yourself.

Now, why i want you to get the

Atma's Impaler

Total Price: 2405
Recipe Price: 825

45 Armor, 18 Critical Strike Chance. UNIQUE Passive: Physical attacks deal an additional 2% of your max Health in damage.

Is because with the next item:

Warmog's Armor

Total Price: 3000
Recipe Price: 980

920 Health, 30 Health Per 5 Sec. Passive: Permanently gain 4.5 health and .15 health regen per 5 sec per minion kill. Champion kills and assists grant 45 health and 1.5 health Regen per 5 sec. Bonuses cap at +450 health, and +15 health Regen per 5.

Your 2% attack of your maximum health, just keeps getting buffed, and with Jax's Equipment Mastery "Jax's prowess with weapons and armor increase his fortitude as he gains equipment. Jax gains 3 health per point of Attack Damage and 2 health per point of Ability Power received from items."

seriously, its a continuing buff that keeps getting bigger, and it doesn't go down. and it buffs your attack, which then buffs your life steal.
GEZUS!

And i just want to say, even though it says this build gives you 3591 health because of what i just said, its possible to get almost 5000 health With all this.

Yes, i don't know much about them, just theres tier 1,2 and 3 runes, and that they can be Primary or secondary. the rest i don't know much about, thats why the rune choise might seem strange. it comes from inexperience.

what i did was look at Searz rune tips what secondary and primary means and more, to choose what kind of runes i would like.
to me i know in the long run, runes will be a good thing to have, but i'm just not there yet to see for myself.
so at this point i would really love input, and im probobly going to copy/paste what you say.

Well as i said before, i borrowed that from BigDowny 's:
"Jax: Dodge, Dodge, Killing spree""

But there's more to that than just borrowing.
I've been told it's always a good thing to get all 3 masteries from the start, so thats one.
and as far as i can see, getting the Leap Strike, to both be able to get away, and get close to the enemy, is a pretty good ability at the start, when you don't have much health or dodge chance to survive on.
Getting the Empowerability next is also something i think is a good choice: since with empower, you can use Leap strike to get close, and empower to hit hard, making it a good 2 sequenced attack. My opinion.

Well, my choise of summoner spells.
Cleanse
Removes all stuns, slows, taunts, fears, snares, sleeps, silences, blinds and immobilizers and reduces the duration of newer applied ones by 65% for 3 seconds.

Cleanse is for when your, lets say, late game, and you have 3-4-5 champions ganging up on you, slowing buffing and whatnot on you, to use cleanse and continue your hitting on them, is usually for me, the best way to either kill the whole bunch, or get away easily.

And
Exhaust
Slows movement speed by 40% and reduces auto/basic attack damage by 70% and ability/item damage by 35% for 3 seconds.

Exhaust is soooo good, when an enemy player is running away.
Beeing able to Leap strike in, use Exhaust so the enemy can't run, and then Empower, is a really good hit combo, and the enemy have a hard time getting away. soooooo good.

Heal
Restores 140 + 20 x level health to your champion and 50% of that to nearby allies.

Heal is ALWAYS good. i just like to have the other spells, to make sure i kill the enemy.
Flash
Teleports your champion to target nearby location.

Flash is good for escaping, chassing, and play around with, it's a many uses summoner spells, that can be used to chase an enemy and kill them, prefereble chasing infront of them (well as long as theres not anyone else that they can smash you over to, something like a turret) its good for escaping because you can get away, close to a tower and get past walls.
Ghost
Your champion ignores unit collision and moves 32% faster for 14 seconds.

Again, a good tool for chasing and or escaping.
Teleport
After 4 seconds, teleports your champion to target friendly minion or turret.

Good for ganking, or getting back to a lane you left really fast.
Of course you shouldn't just run after enemy champions when you teleport back to the lane. don't feed them.

Hargis2010

not to be over critical but you should never get flat health with Jax his passive gives you health for making him more potent(only health i would maybe reccomend would be frozen mallet or rylai's). Secondly, rather trivial compared to the first; your masteries. ALL of his abilities deal magic damage.. so get archaic knowledge. Also, jax's main weakness is casters(magic damage) so max resistance instead of hardiness.

skill sequence: max leap strike sooner, with 5 ranks into it the cooldown becomes reletivly short compared to the 10+ sec at lower levels.

So with that said, i agree somewhat with your build with the exception to warmog's. If you are dead set on health as Jax, try getting both mallet and rylai's, its actually quite good.

Just trying to be constructive :/

---

So here Hargis mentions something i haven't thought of. well, i put it up as you can see, so take from it what you could need.
personally im going to try it out. see if it works better for me.

Thats it for me.

what im going to do is put up a chapter with "Peoples Input" and just copy/paste.

Well if any input is coming in.
This is my first build, but i hope its ok.
Im not that high of a lvl in lol, lvl 14 right now? But this is how i play jax, and i think he's awesome.
hope you like him just as much as i do.

P.s you can also try use The Black Cleaver instead of Frozen Mallet