Which magazine are you referring to? The debug / start log for the current session can be cleared using the "Clear" button on the top panel, but all sessions are stored in the "Log Navigator".
Saved logs from past run / build / whatever sessions are stored as part of your “received data” for each project. To view this, open the Organizer window and select the Projects tab. Select the appropriate project from the list on the left, then where it shows the derived data folder, click the arrow to the right of the path to see it in Finder. You can trim the logs in the "Logs" subfolder manually if you want (keeping more recent logs), or just click the "Delete ..." button in the far right corner of the path in the "Organizer" window. Doing this will destroy the logs, its index, and output its assembly at a time.