It depends. Are you trying to protect the trademark? If that's the case, then I probably would if you think it is worth protecting.
If it is a generic logo, it might not be able to be trademarked anyway, so if that's the case, then no.
If it is a logo that could really only be used for your business, it is doubtful it will be copied. If this is the case, I wouldn't still.
If you think you can file for a trademark yourself, then perhaps that route would save you money, or you can rely solely on the common law trademark to protect your logo. If you drew it yourself, it may also be copyrighted, and thus automatically protected when you drew it.
No matter what you choose, it can't hurt to find a trademark attorney who does free consultations to see what he or she thinks.