No - a full backup does not truncate the log. The log and database backups are somewhat independant. When you backup the log, it will contain all the log entries from the last log backup regardless of when the last full backup was taken. For example, if you backup your log at 8AM, then do a full backup at 9AM, and 10AM and do another log backup at noon, you can restore to 11AM by using either full backup and the noon log backup (the noon log backup contains all the log entries from 8AM to noon). In order to do a point-in-time recovery, you must be doing log backups as well as full backups. If you only do full backups and truncate/shrink your log, you WILL NOT BE ABLE TO RESTORE TO A POINT IN TIME except for the point in time where your full backup was taken. Doing truncate/shrink on your log file doesn't give you any benefit different from having the DB in Simple mode, you lose all the log information.
Good luck - please post any additional questions you have as it appears your current setup will not allow you to restore as you had hoped.