How to backup Console logs?
Console.app's logs are stored in the following locations:

/var/log
/Library/Logs
~/Library/Logs


To back up all the logs, simply copy the contents of these three folders to another location.

To back up a single log, look in Console.app for the log that you want to back up, select it in the sidebar, then press ⌘R (or right-click and select Reveal in Finder). This file can then be copied to another location.