If I want to trademark a phrase and an image logo that includes the same phrase; will trademarking just the image also protect the non-stylized phrase?

Whatever is in your TM application, which should reflect what you are actually using as your trademark, is what will be protected. So, if you present, as your trademark, the logo, which includes the phrase, both should be protected.

I agree with Mr. Weinrib. And in reference to the question of costs, each application can contain only one mark. So you can have one application that contains both the phrase and the logo together, assuming you use the mark as such, or you can have two applications with one covering the slogan and the other covering the image (assuming the two components are separable and not intertwined). Each application usually requires a filing fee to the trademark office and if using an attorney, the attorney may require a separate fee for each application or perhaps could charge one fee for both depending upon the arrangement you work out. Consequently, you could potentially be doubling the cost when applying for both but cost should not be the main consideration. Your main consideration should be what is the most effective and necessary way to protect your particular mark for the manner in which your particular business uses the mark and where your business and the mark may be used.
The location of the business and mark usage is critical because if in the U.S., you can obtain trademark rights merely by using the mark in U.S. Commerce without any registration; but in many other countries a mark needs to be registered in order to have rights in it; and some countries may not even register or protect trademark rights in a slogan.
While you may be trying to avoid incurring the cost of hiring an attorney, you really should consult with a trademark attorney to review your particular situation because trademark law is very fact-specific and subjective and hiring an attorney could probably ultimately save you money. A trademark attorney can advise you of the most effective way to protect your mark(s) for the least amount of money.
* Please see my bio for an important disclaimer.