Have you ever made a logo or had a designer make a logo? You do not make a logo for a client in Photoshop. That’s just unprofessional and not how it’s done. If you make a logo in Photoshop, it’s just stuck at that size. Logos are always delivered in vector format.
Besides, it’s not like there’s really much to the Arcadian logo anyway. Just install the Dancing Script font and write out your new text in Photoshop. Judging from your question and response, I wouldn’t advise actually buying Illustrator if you don’t already have it and have a grasp on these concepts. You don’t need to learn a new software just to write some text out in that font.