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.