Logic Pro disappears after download
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.