Hi Johnny,
Since you reformatted your computer, the file (/Library/Preferences/com.apple.musicapps.InstallerHelperTool.plist) that keeps an encrypted reference to the location where you relocated your Sound Library to is now gone.
However, as Logic scans all available drives for Sound Library content (which will always be in "Logic" and "Garageband" folders under /Library/Application Support in whichever drive the Sound Library was relocated to), this is how it is able to tell that you currently have Sound Library content in that "Logic Library" drive of yours.
So, if you want Logic to reuse the Sound Library content you already had relocated to your "Logic Library" drive, you'll have to do the following:
Quit Logic
Go to your "Logic Library" drive, locate the "Library" folder that will be at the root of the drive, and rename it to something different (for example "_Library").
Launch Logic and go to Logic Pro > Sound Library > Relocate Sound Library... You should now see your "Logic Library" drive a selectable drive to relocate the Sound Library to.
Click Relocate
Once it's done relocating whichever content you currently have on your system drive (assuming you did not download everything to your system drive yet), quit Logic.
Go to your "Logic Library" drive and trash the "Library" folder that resulted from the relocation you just performed, then rename the "_Library" folder back to "Library"
Next time you launch Logic, it should now use the Sound Library content you where using before you reformatted. You can verify this by going to Relocate Sound Library... where you should see your "Logic Library" drive as being the "Current Sound Library files location".
J.