Full backups do not contain the transaction logs. That's a myth. A full backup has a complete copy of the database including all completed transactions up to the point of the backup. Any incomplete transactions are cleaned up as part of the restore process.

The thing is, a full backup, depending on the size of your system, can take quite a while and will use some system resources. So you can't run a full backup every 1/2 hour on most production systems. Instead, other backups are used in combination to protect the data in a more granular fashion. That's what log backups are for. They backup just the transaction log. This allows for very fast, lightweight backups, and, as was already mentioned, the ability to restore to a specific point in time. A full backup can only be restored to when it occurred. Logs can be restored to specific moments.

Backups are not a technology problem. They're a business problem. Talk to your business and find out exactly how much time they can afford to lose on the database. If that number is small, you're going to need to work on setting up a recovery mechanism that covers the business adequately.