Open the App Store on your wife's computer and log in with the Apple ID you have previously bought Logic Pro X from the App Store. Now, search for Logic Pro X and you should see the Install button.
After the installation and download of the files you need, you can log out of your Apple ID account. Whenever there is an update for Logic, the App Store you remind you, and it asks you to log in with that Apple ID again if you want to download any future updates.
And to make sure. You wife is not allowed to use Logic on that computer while you work at the same time in Logic on your computer. Install the app on as many computers you have access to, but only use one at a time.
Hope that helps
Edgar Rothermich - LogicProGEM.com
(Author of the "Graphically Enhanced Manuals")
'I may receive some form of compensation, financial or otherwise, from my recommendation or link.'