[4.3] A Baron Lane Jax Guide!

STARTING ITEMS

• Long Sword is the offensive option. Build it when you are confident.
• Ruby Crystal is the defensive option. Build it when you are not confident or to play safe. Builds into Divine Sunderer and boots.

CORE ITEMS

You always need a sheen item, so choose between Divine Sunderer or Trinity Force . I always prefer Divine Sunderer , unless the enemies are all squishy then get Trinity Force but even then, Divine Sunderer remains a good option in such situations.

BOOTS & ENCHANTMENTS

• Plated Steelcaps and Mercury's Treads are always the boots for Jax as they offer the matchup-specific defensive stat that Jax needs.
• Ionian Boots of Lucidity is also viable for the ability haste, but you become somewhat squishier.

For the Enchantments, It's free of your choice as I think it depends more on your playstyle. But I alternate between Gargoyle, Stasis, and Quicksilver.

SITUATIONAL ITEMS

• Death's Dance is good if you need armor.
• Blade of the Ruined King good sustained damage item, popularly bought, but not mandatory.
• Sterak's Gage Makes Jax can survive bursts while offering a huge amount of AD.
• Guardian Angel . Revive and armor, nice. Also don't forget to replace it with other items while on cooldown.
• Black Cleaver is used to shred armor. Good when facing many champions that build a lot of armor, and the ability haste can be juicy for Jax too.
• Wit's End is your go-to item against AP threats.
• Riftmaker is usable for getting omnivamp, something that Jax likes because he does both physical and magic damage.
• Nashor's Tooth offers a huge amount of attack speed and ability haste, and also some more magic damage on your auto-attacks similar to Wit's End .
• Frozen Heart is an alternative option to death's dance if you need more armor, losing some AD but in return, you have more mana, armor, and ability haste.
• Randuin's Omen against 2 or more crit threats, or if one of them is really fed.
• [chempunk chainsword] for anti-heal.
• Thornmail for anti-heal for solely AD threats, can be bought instead of Chempunk if they have fed ADC for example.
• Force of Nature is best second MR item if the enemies are mostly AP.
• Maw of Malmortius is also a contender for force of nature. This item is more offensive than Force of Nature , build it when you are more confident.
• Hullbreaker for taking down turrets early while also boosting your 1v1 power with the resistance it provides.
• [amaranth's twinguard] allows you to become a supertank in the late game while also still dealing a lot of damage.

OTHERS: 

• Sunfire Aegis is niche. Mostly bought it for the HP and the damage it provides.
• Spirit Visage for teammates who heal a lot and if the enemies are all mostly AP.
• Mantle of the Twelfth Hour is an alternative for Sterak's Gage if you want to be more defensive and is still a good purchase against hybrid compositions if you don't know what to buy next.
• Abyssal Mask keeps your mana full and is purchased against AP threats. But still not as notable as other options here.
• Crystalline Reflector is a magic armor item, that surprisingly offers a good amount of defense but still, compared to Death's Dance and Frozen Heart, this item doesn't offer Jax any ability haste at all.