League can sometimes get stuck as "active" program in the Task Manager, preventing you from running another instance of it. The issue is basically the same where there's only League of Legends logo on your screen and it infinitely loading if you mouse over it. Killing any League tasks and starting the game again fixes this problem (even restarting a computer) because it unloads all RAM.
Starting the program as "admin" fixes a lot of other issues, if for some reason the account you're using (likely Guest) to play a certain game or start a program, was not given the permission to modify certain files. Right-clicking on the shortcut and setting the option "Run as Administrator" should fix this.
It's also likely that a "stealth" setting in your Antivirus simply stopped League's launcher from running and has moved it in Quarantine, blocking it permanently as it's on a blacklist of programs that aren't allowed to run. "Stealth" setting means that the Antivirus is set up in such a way to automatically block certain processes and not bother the user with a notification that a suspicious program is trying to modify files. This issue happened to me as my 360 Internet Security actually removed League's logo file, preventing it from running because it did not have all the required files.
It was a pain in the butt of figuring this out as i'd never suspect it, if i wasn't notified of such an action beforehand.
- League of Legends Logs Files
By going into your League's installation folder, locate the folder "Logs" > "LeagueClient Logs". Look for the most recent time a log was created (the last time you tried to start League) because to me, it seems that the client itself won't launch (not passing the logo).
In that specific text document, there's certainly a phase where it says "ERROR" on the left side instead of "OKAY", which could indicate the possible issue. However, if you're trying to demistify log files, this is a job for Customer Support.
