Hi there, strictly speaking, buying a used MacBook Air does not entitle you to ownership of Logic license even if the software is already installed and ready to use. You may still use this copy of Logic, BUT this could potentially open a can of worms from the legal point of view. Especially if you are making money from that or produce music that sells. If it doesn’t generate revenue, then fine, I guess, who cares, right? Still, I think the simplest way to solve your issue is to buy the app using your unique Apple ID. Then you could install it on as many Macs as is allowed under terms of the license agreement (i.e. two machines you own, at any given time, if I’m not mistaken).
Haven’t heard of “adopting” the app, as indicated in posts above. Perhaps, this is a procedure for apps pre-installed on new Macs which are sold bundled with software as a marketing gimmick / hidden discount (i.e. buy the hardware and get software for free). But if that’s the case, you’d have to call Apple’s support line to find out whether that particular MacBook Air had been sold together with software. And if it wasn’t sold like that, it’s unlikely you could “adopt” the app, because the license would belong to Air’s previous owner. Apple doesn’t allow for the transfer of its app licenses, as someone had pointed out in another thread here.