There are two storage groups backup up by Backup Exec. They are set up as separate jobs.
Both have logged successful completions but on one of them, Exchange is not deleting the old logs.
On Exchange I checked:
1. With Powershell:
get-mailbox database -server 'server' -status | format-table name,storagegroup,lastfullbackup
This shows the last full backup time is 7/24/2016, same as reported by Backup Exec
2. With cmd:
vssadmin list writers
This shows no errors with any of the writers. All are in State  Stable except for 'Microsoft Exchange Writer' and 'Shadow Copy Optimization Write' which are in State  Waiting for completion.
There is a backup for the problematic Storage group running now (after I made the changes listed below in the Backup Exec description), so I think this is normal. Please correct me if it is not.
3. Event Viewer shows ID 9827 logged after every successful backup.
Interestingly, I do not see it logged for the Storage Group in which the logs are being flushed.
Backup Exec Settings
I compared the settings set with the other job that is working properly. Everything looked proper except for the #2 and #3.
1. Full - Database & Logs (flush committed logs) - This was already set
2. Verify after backup completes: Yes
changed from 'No'
3. Let Backup Exec automatically choose the best copy to back up (recommended)
changed from 'Backup from Active Copy only'
I am running the backup job again after changing the "High Availability Settings" to the recommended value.
Is it possible that backing up from the Active database and not verifying after completion is the cause of the logs not being flushed?
What other logs should I check to see if the problem lies elsewhere?