how can I check system log older than one month

Log files are deleted after a given amount of time unless backed up. You can view the logs remaining on the system by choosing Go to Folder from the Finder's Go menu and providing /var/log/ as the path.

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Unless 'vuse' runs its own logs there's no way to tell whether it was running, or what it was doing at a specific time. The OS doesn't automatically log every application's actions all the time. Can you imagine the size of the logs files if it did?

You might be able to work out if the machine was on at the time by looking at the system logs, as long as they cover the time period you're interested in. Earlier than that (typically a month) about the only thing that's left is the last log (open Terminal.app and type: last ) which will give you a list of recent logins, shutdowns and reboots from which you might be able to extrapolate the data you're after.