I'm not sure what a script logo is.
I would draw a circle with the elipse tool. Then select Type on a Path click on the left of the circle. If you want a half circle, or just an arc you'll need to pull the scissors out and do some cutting. Once you get your logo onto the path, you'll need to replay around with it to get it just right.
An easy tip for centering text on an arc is to use spacebar, if your text flows farther than originally intended and looks lopsided and you can not go back any further via backspace. Then take the rotate tool, center it on your elipse and gently rotate left until you've made a nice centered arc.
