I recently got a new (old) MacBook Air, to help record on the road, and whenever I try to download Logic Pro X, it finishes the download (~800mb from IIRC) but then disappears from the Launchpad. I managed to successfully download and install Final Cut and MainStage with no problems, but Logic just won't stay there. I've got about 100GB of free space and I've tried searching through Finder for the app but to no avail.
I had the same problem.
The reason it's happening is that Apple is having a problem with their Australian download server. You can get around it by using a VPN to connect to somewhere in America, and then downloading Logic via the App Store as usual.
Alternatively, if you have Logic on another computer, you can copy it from the Applications folder - just drag and drop onto a USB thumb drive. Then drag it from the thumb drive onto the Applications folder of your new computer.