In general any creative work is automatically copyrighted once it's fixed in a tangible form e.g a JPG file, so you don't really need to do anything special unless you think you will have to take legal action against someone using your logo later on. If you think that would be the case (and if you are in the US) you need to register it with the US copyright office. This will give you a public record that can be used as proof, and you'll be able to sue someone for damages related to copyright infringement. This whole process should cost you a $45 filing fee. See How to register a visual work. If you are in any other country you need to visit the copyright office/department for your country and find out what the procedure is.
No you cannot patent it. GoHalo3 is correct with regard to copyright and trademark. The cheaper way is copyright and also is less burdensome if your logo is being used for a business that hasn't really conducted business. Trademark filing is much more complicated. You don't need to copyright your image but if it's of that importance to you, it may make sense.