It is completely random what logs or other items you will obtain from Logging. It is possible to get multiples of the same log from a Logging Point.
You cannot purchase the logs and trade them to Ramua. She will refuse to take them. They must be logged and you will see the message pertaining to that.
When you successfully cut off a log for the first time of a kind you need, you will see the message: "<Your name> recorded survey data for a <log name> on the Timber Survey Checklist."
Subsequent logging of a log you've already recorded does nothing extra.
Return to Ramua and trade her your five logs to complete the quest.
Ramua will confirm the logs and allow you to keep them.
Game Description
Client: Ramua (Carpenter's Guild, Northern San d'Oria) Summary:
The overworked Ramua asks for your aid in collecting timber sample for a guild survey that she's been too busy to conduct herself.