*UPDATE*

I will post how to jungle as Jax in this section in case your team does not have a jungler.

I take Exhaust and Smite (24/6/0)

I start with Cloth Armor and 5 Health Potions. If you're up against a balanced or magical team, upgrade the cloth armor into Guardian Angel later or sell it later.

I get blue buff at level 1 with my teammates' help. From there I go to dogs, wraiths, lizard, and golems (in that order).

Welcome to my Jax guide. I'm not here to brag. I just want to introduce my fellow Jax players to my style of playing Jax. Before I begin, though, I'd like to say a few things. If you disagree with something inside this guide, I encourage you to point it out so I can fix it. However, you don't have to rate my guide down. I think the only scenarios in which rating a guide down is appropriate is when there is something blatantly wrong with it. For example, an Ashe build that doesn't have flash or ghost in it. Even then, you could point that out instead of rating the build down.

I'm a longtime Jax player and I'm good with him. I'm a tricky Jax to deal with in the laning phase even though my early game is always essentially a gimped one. I don't just play Jax though. I also play Jarvan IV who is my second main. Like I said before, I'm not bragging, but just so you know that I know what I'm talking about, I have more than 300 games played as Jax in the Ranked Solo Q. My top rating is 1501 too.

OK so I am going to stop typing like a professional writer now because there is a LOT of writing to do and I don't feel like being 100 percent grammatically correct every step of the way cause this is going to take a long time to write as it is. Anyways, you probably have noticed that there are 3 different Jax builds. I will explain why here.
They are arranged in order of Most Used to Least Used (the first one being the most used and the third one being the least used). I call the first build the Standard build. You use it when the enemy team's damage is balanced. I call the second one the Anti-Magic build. You use it when the enemy team's damage output consists of mostly magic damage. I call the third one the Anti-AD build. You use it when the enemy team's damage output consists of mostly Attack Damage.

Before I get into the details inside the three builds, I'm going to use this section to talk about how to accurately play Jax in-game.

When duo-laning, Jax is an excellent zoner when in the brush, because his Leap Strike - Empower combo enables heavy harassment with a lot of pressure. This can even push two champions away from experience range while allowing a teammate to farm.

It is recommended to take minion aggro to build up your Counter Strike and then Empower and then Leap Strike, activating Counter Strike mid-air to ensure that you can make it out of there without getting focused down and or punished for jumping on them.

It is also recommended that there is a bush of some sort around the enemy that you are leaping to so that you have a place to hide out in while waiting for your cool downs.

Jax can use Leap Strike to leap into the fray of a fight or make a quick escape when necessary due to the fact that Jax can Leap Strike towards a friendly unit.

The best strategy concerning Empower is to melee strike, then activate the buff immediately after. Empower resets the attack timer, so activating after an auto attack will initiate three quick strikes, one auto attack, Em-power's attack, then another attack rapid fire. This, properly done, will vastly increase damage output over time.

Be patient with Jax. Often the opposing team will focus you down due to your high DPS output. If you are having a bad game, just focus on gaining gold through minion kills, assists, and jungling. Once you get the items you need, you will be much harder to kill.

Yes, I did take these from the Jax page on the LoL Wikipedia. Speaking of the Jax page on the LoL Wikipedia, you can go there to get more information on Jax like EXACT statistics.
Jax

BUY THEM! Having a ward as Jax is almost like having flash. Step one: place ward over wall. Step two: jump on it!

The Mercury's Treads give you CC reduction which is important. Ninja Tabi has dodge and dodge is useful on Jax, but guess what?! CC reduction is a lot more useful. I usually get the Pickaxe and the Amplifying Tomes before the life steal or spell vamp because the flat damage statistics give me the HP I need to stay strong in the lane. I also sometimes buy an early Ruby Crystal. I put the G-blade on slot 5 because that's the number on my keyboard I am most comfortable with reaching out to. Well I think this section needs to be divided into more sections.

The armor penetration makes Jax a lot easier to last hit with. Although Jax does magic damage, he is still a right clicker, and you don't ever go wrong when giving armor penetration runes to a right clicker. Since you probably won't be buying Ninja Tabi very often, you must have dodge seals, or else your E will proc too rarely.

So you noticed (or should have) I don't have the Evasion mastery. Well, let me tell you up front, relying on dodge is a mistake for many reasons. #1 dodge is a percentage, you are gambling with it #2 Jax can't dodge spells.

If Jax's Q wasn't so damn useful, I would for sure be using ghost or flash.

It used to be a gigantic spider web I made. There were so many items in the web. Eventually, I realized which ones were superior. I didn't even get it right after that. I was still using inferior item choices. Here are some notes I made when I was done updating the build:

*changed Thornmail to Sunfire Cape
*changed Sunfire Cape to Atma's Impaler
*changed Force of Nature to Abyssal Scepter

So you see this build went through a long crafting process. Now you might be wondering why I am NOT using a RAGE BLADE. Well, flat damages are over rated, when you are in RIGHT CLICK TARGET mode, it is more efficient to use critical strike chance. If you do make it to the Infinity Edge before the game ends, you will see just how much of a monster this item build makes Jax become.

Only PRACTICE can! You can't be told how to play and then the next time you play you dominate in a way you didn't before. Over time, League of Legends instinct will guide you to victory. Let me tell you how a typical ranked game as Jax goes for me. I almost always start with boots and 3 Health Potions.
My teammates and I head into our jungle to protect it. If we decide that our team is stronger than the enemy team at level 1, we invade the enemy's jungle. I usually solo top or lane bot with a support. Early game revolves around ganks and farming and more ganks and more farming and Dragon. Middle game revolves around pushing and farming and Dragon and Baron. If you reach End Game, make sure you stick to your teammates. End Game revolves around team fights and Baron.

Farm early game
Dominate middle and late game

That's how I play Jax ^.^

of course if you are horrible at last hits, use flat AD runes in the hopes of getting a kill or two early game.

And if you are just a casual player, do whatever you want I guess.
If you want "Jax lessons" just send me a friend request, my user is The Skilled Noob
And if you want more quality guides like this one, my profile is not a good place to look, cause I made and published most of my other guides back when I was a newbie. I got a quadra kill at 1600 elo with an older version of the build, so he-he you can't say the build doesn't work. I was only able to play at 1600 elo because I duo Q'd with TheAnomaly though. I've also gotten two penta kills at 1300-1400. But of course if you need proof here it is

1300-1400 elo:

http://i54.tinypic.com/5vnk28.jpg
http://i53.tinypic.com/21cv38o.jpg

1600 elo:

http://i54.tinypic.com/21ovmmw.jpg

As I said I don't brag I am just showcasing what the build is capable of.

lol this picture I have here is funny:
http://i55.tinypic.com/nyigao.jpg
I laugh every time I look at that one lol