Before your beavers can begin providing logs, you'll need to make an Isolated Securement Unit for them first. You can find it under the Facility tab of the Build menu, provided you've unlocked its associated node in the Memetics screen. Place the Isolated Securement Unit anywhere on your base, then deposit your Logging Beaver inside it. The beaver will automatically start logging whenever it has enough energy to do so, and you can claim all the Logs it got by interacting with its Securement Unit...