Hey I just was wondering if you don't log out but close your browser, do you still appear to be online when you see the names at the bottom of the Forums for users online?

I have always logged out when I am done looking at the site, but am wondering if I don't will my name still appear as I am online?

Yes, you still appear to be online for half an hour from the second of your last activity on the site, unless you logout.

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Any reason to log out besides that?

Other people using your computer can go to this site while you're still logged in, that's if you don't log out. Other than this, I don't see any harm for not logging out.