Can I purge transaction logs without taking a backup?

Yes, you can purge transaction logs without a backup, but you might as well ask, "Can I take out my own appendix?" You can certainly do it, but it's not recommended. Recall that the transaction logs keep a copy of transactions posted to your Information Stores (ISs). If you're performing regular full backups of the IS, in theory you can restore a backup to recover the data as of the time you took the backup—assuming that the backup isn't damaged, corrupted, or otherwise unavailable.

The logs are a safety measure; in the event of a failure, you can use them to replay any transactions that might have occurred after you made the backup. Therefore, purging the logs makes it impossible to replay these transactions. If you're making full or incremental backups with Exchange-aware backup software, the software automatically purges these logs when the backup is complete. If the software isn't purging the logs, your backup process has a problem that you should investigate posthaste, beginning with the backup logs.