What type of maintenance jobs do you have in place? Do any truncate the transaction log or change the recovery model to SIMPLE (even if only temporarily)? Something is likely breaking the log chain, and you need to figure out what.
There isn't a way to fix the LSNs if the chain is broken. You need to figure out what is causing it, fix it and then start a new chain.
It saves the backup in a specific location, and gives it a consistent filename including DB Name, date & time. That stops anyone just making a "quick backup" to some weird location - including then moving the file to their C: drive "just in case" - all of which is useless when it comes trying to recover a system when there is a real life disaster
Having said that, you can query the Backup History Table in MSDB to get a list of the actual backups that have been made, and that would have listed the missing ones that you knew nothing about