How does trademark law protect a logo?

Yucel and Steven have pretty well answered your question and their answer could be summed up with the classic economic cliche, "you get what you pay for." The more registrations you pay for to cover the various elements of your mark in the various manners in which you use your mark, the more protection you should have for your mark. Also, each registration is a business asset so technically your business should be gaining some additional value with each trademark or service mark registration it procures.
It should also be noted that any mark that you use with your business in the U.S. is accruing "common law" trademark rights which can be enforced even if not registered. However, the registration adds enforcement power to your exclusive right to use the mark in that the mark can be enforced throughout the entire U.S. and proof of your ownership and validity of the mark is not necessary for a mark that is registered. So if you don't register a portion of your mark, your mark may still be protected under the law and you can still attempt to stop others from using a mark similar to that portion of your mark but it could be more difficult and costly to do so.
(see my disclaimer in my bio)