Can I find out who left an anonymous comment?
No, you cannot definiteively find out who left an anonymous comment. The term anonymous means that the identity of the author is withheld
If you do not wish to ban anonymous commenting altogether, you may choose to track the IP addresses of anonymous posters. The IP address will not provide definitive information as to the identify of the poster, but will allow you to identify the poster's Internet service provider (ISP). To determine the IP addresses of anonymous poster, you must enable Comment IP Logging in your journal for at least "Anonymous posters only". This will not display the IP addresses for comments made before this option was enabled.
FURTHER READING
How do I control who can post comments in my journal?
http://www.livejournal.com/support/faqbrowse.bml?faqid=23
What is Comment IP Logging? What do you mean, "this user is logging my IP address"?
http://www.livejournal.com/support/faqbrowse.bml?faqid=66
What should I do if I'm being harassed by an anonymous commenter?
http://www.livejournal.com/support/faqbrowse.bml?faqid=109
EDIT
It has been pointed out that when trimmed down to only what I intended to write about, it is a very meager and useless FAQ. Sorry to waste your time or get your hopes up!