Probably not. But you more than likely can modify it to work on your 10" Logan lathe. Offer it for trade for a Logan brand steady rest and see what surfaces. I'm partial to Sheldon lathes and hate to see the steady rest get modified for another brand of lathe. I'm guilty of it myself. I've been very fortunate to find the correct SR for my Sheldon lathes. But not so lucky for my 14" Rockwell lathe. It will get a modified Sheldon steady rest that some one modified to use on a mill table. My Sheldon's are a 13" and 15" early R series lathes.
Agreed, offer it up to the folks on the Sheldon Yahoo group. You should find a buyer quickly. Use the funds to buy the correct Logan SR. I've made sure my lathes had the correct SR before buying. My current Sheldon was mint and well tooled with the the original SR. But it didn't have a follower rest. I finally found someone on the Sheldon Yahoo group that had an extra.
I just bought a 1952 Sheldon 11" by 36". I have not yet checked the steady, but did have occasion to use the follow a couple days ago.
Getting it out, it looked like it should fit the bolt holes in the right side of the carriage and did, once I has chased 64 years of hardened
dirt out of the boltholes with a tap and air hose, and had purchased some grade 8 bolts since they were missing. I found that the rear
screw could not be backed out for maximum diameter without hitting the crossfeed carriage. I had to beat the jaws loose with a hammer,
as they were frozen into place by a fit so tight it couldn't even rust!
But the biggest surprise was that the jaws did not contact the workpiece on the center of the jaw. I am inclined to think this follower
belongs on a 10", Perhaps someone has a 10" and your follower has a similar miss-match and we could trade?