Make sure you're looking in the 'air' folder inside your League of Legends folder. The air folder has everything for the startup client, so you should see things like "Adobe Air," "LolClient.exe," etc. The log files are created any time the client starts up, so the log folder should definitely be there if you've started up the client anytime recently.
Full path is:
]C:\Riot Games\League of Legends\air\logs
You'll see a bunch of files that have names like