How does trademark law protect a logo?

It depends upon how you use the logo, with/without company name and/or the slogan. For example, GENERAL ELECTRIC has the stylized GE logo which is sometimes used with or without the company name GENERAL ELECTRIC. The same thing goes for slogan text - for example, they may use GE INNOVATION AT WORK - all of these uses and variations need to be protected by separate TM registrations if protection for each and every one is sought - GENERAL ELECTRIC is commonly referred to as a "house mark" because that is the main company mark, whereas the other marks are additional trademarks protected by General Electric because they want to protect the different marks - the logo, the slogans, and the like.

Sometimes applicants find solutions (!) to reduce the cost of registration such as GE GENERAL ELECTRIC INNOVATION AT WORK. Instead of proceeding three applications they wish to solve the problem with one application. It reduces the total cost of three registrations but the owner was going to use three factors separately and it may cause a problem. This makes the protective power of the registration weakened. Infringers use the actual usage of the trademark. After registering GE GENERAL ELECTRIC INNOVATION AT WORK, if the owner uses just GE or GENERAL ELECTRIC he makes a differentiation from the registered trademark. If an infringer wants to move closer to the sign and he starts to use GENERALLY ELECTRIC, what shall be done against him? The owner of the trademark additionally needs may be a help from unfair competition law to stand by trademark registration. I always strongly advice to register the trademark as you use it. If there are more than one, then may be the filings will be more. Adding side factors to an application because of different motivations may damage the distinctiveness of the trademark or logo or slogan.

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)

Check out the cartoon we published on combined trademarks: http://trademarkfactory.ca/cartoon/combined_trademarks_file_separately_or_together

This should answer your question about protecting separate elements of a trademark that contains multiple such elements.