Firstly, are you using an Exchange aware backup application and is it set up correctly? Check you're specifically doing an Exchange aware backup job (as opposed to some shadow copy of the .edb and log files) and that all license keys and/or agents required are correctly installed and configured.
Secondly, assuming the application is doing an Exchange aware backup - are the backups actually completing successfully? Your backup application should be able to provide you with information as to whether the backup completed, failed or skipped some things - that would be a good log to interrogate. The Exchange server itself will also log events relating to ESE backups - they might be worth checking out in this case.
Lastly, it may sound like a dumb thing to say, but are the log files you see definitely associated with a database that you have just backed up? Is there any possibility these log files belong to another database that is either not being backed up, has backups failing, or the database itself has simply been deleted and the logs left in place.