You are using both DPM and native SQL at the same time to do backups. If you want DPM to do the backups (as you should) stop running SQL backups. You cannot run backups (unless it is a copy-only backup which doesn't truncate the log) from multiple sources. They will both be broken (except the full backups) because each log backup will mark the log LSN and truncate the log - as it should). This is why you are getting broken log chains.