You want the wave near your turret always in this matchup so he cannot jump on you. Try not to push the wave when you are farming or poking with q.

Q>E Max If you want to play for only q poke onto Jax especially if he is not running a sustain setup with Second Wind or Biscuit Delivery and Doran’s Shield, you can build the ap components of your Nashor’s tooth first and run Doran’s Ring for mana regeneration. When you get to lane ward the bush closest to your turret and walk with your wave to ensure Jax cannot pull the wave or look to zone you from the wave early. From level two onwards start respecting Jax’s leap strike range. If he jumps on you pre six, you have no way to trade back and will be losing health for nothing.

When Jax uses leap strike on you, immediately use your q on him and try to get of out range of his counterstrike so he cannot stun you. Depending on how many points you have in your q you may or may not have to use your w to have enough movement speed to get out in time. Post six, you can start autoing him after you get out of his range and take very heavy trades.