I want to start a company and I have a logo in mind. For example I want to put "Snappy Duck" as my brand logo on all my clothing, pants and shirts. I also want to copyright my desings for shirts.

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http://www.uspto.gov/trademarks/index.jsp

Logo copyrights are sometimes tricky. Copyright law states that there must be "original authorship" in order for a work to be copyrightable, and words, letters, and short phrases are not copyrightable. By this logic, if your logo contains a significant amount of original material (such as a picture, for example), you can copyright that material -- but not any accompanying word or short phrase. (And simply adding artistic treatment to text is not considered original enough.) There's a logo copyright test linked below that you can check out to get a better idea if your logo can be copyrighted. Trademarking, however, might be more along the lines of what you want -- while copyright law dictated that no one can use your image without your permission, trademark law means that no one can use your logo for personal gain. The protection is about the association of your logo to your business, not just the image itself. Good luck to you!