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To me, the real question is, "exactly what do you mean by 'the background' "? Do you mean the sky, or do you mean something else? Personally, I would consider the sky to be part of the image, not part of the background.
Tom M
What are you referring to when you say "logo?" The text? The anime art? Everything but the sky?
The best way to do it is probably using a selection tool. The problem is that the lettering has an outer glow, and it will be difficult to select that portion, which will make the lettering look cut up and uneven.