How is it cost a new logic board for a macbook air 13 inch?

You need to contact Apple or an AASP to get an estimate for the repair since prices vary by region and you have to consider differing labor rates & taxes for each region. Plus if accidental damage is found the repair prices goes up a lot more. As a ballpark figure it could cost around $500US.

Mostly even more...

In south Florida it was going to be $550 for a new logic board for my 2018 13" MacBook Pro.