I included a bunch of logos to help you brainstorm. I'm open to different ideas, but my instinct is to have "prose" in a large sized font, and "media" in much smaller font underneath it (on the right side). Perhaps "prose" in lower case and "MEDIA" in upper-case, or other way around? You're the expert. Open to any options.
Since the name itself is already serious, the logo font might be a bit more fun / warm / inviting.
I love the color blue, and the look and feel of the website is somewhat inspired by freshbooks.com.
The most important trait to convey is "quality" (high-quality writing), so it might be possible to make it look a bit literary/distinguished and even play on the writing theme (e.g., with a pen).
These are all just ideas. I'd love you to use your creativity and see what cool ideas you can come up with!
(And due to the nature of my business and client base, I might also be a good source for referrals.)