No, separate applications are required. One application would be directed to the words and the other to the logo. You will typically see the first type of application called a “standard character” or “word mark” application and the other a “stylized” or “design mark” application. Depending upon how you use your mark, it might also make sense to apply to register the design element of your logo alone – that is, if you use it separately and it has independent commercial value.
You obviously have to be practical because it can get expensive to file multiple applications since the Trademark Office charges a per class, per application filing fee. However, most law firms (The Kinder Law Group included) will give you a big break on total fees if you file multiple applications at the same time. Also, there are ways to file your applications to get more “bang for your buck.”
The Kinder Law Group happily provides free consultations if you have any questions.