Exhaust will be fine because it adds crowd control, that works well with the E of Jax for good rotation of cc. Exhaust + E nullify completely opponent's burst, Jax was strong in 1v1, now he's really godlike.

Ghost will be fine because Jax has already gap closer (Q) as strong as flash, ghost will make easier to make proper E. furthermore, ghost will make easier to escape in addition to Q. I like flash too for fast engage in addition to mobility boots (with alacrity), Q and E but ghost will be better for gank other lanes and 1v1.

Crystalline flask is really good one item, you got free sustain after first back you don't need to buy other potions, just own Vampiric scepter. For you, the most important is management of the lane (ward, freeze, trade, push, back etc.)

Finish your Bilgewater scepter and go on Trinity force (Phage -> Sheen -> Zeal)

In mid/late game you will roam then Mobility boots will be fine for fast run and come back to the lane for depush. Do not hesitate to enchant your boots with Alacrity at this moment of the game it will be an enormous asset. So finish your Hextech gunblade. Now no one can trade you in 1v1 you have from then on the best potential of split push of league of legends.

Now you can buy magic resistance for go to Wit's end and Spirit visage (without Exhaust you are really squishy vs AP Burst)

Finally, end by Guinsoo + Elixir of Wrath and go owned this f*cking pentakill