The “Logan” verse you are referring to is in the Surry County version of this song, which is referred to as “Susananna Gal”. The verse reads:

Hitch old Logan in the front and Susan in behind him
I’m going down that rocky road to see that gal of mine.

Logan and Susan are mules that you hitch to a cart. In the Roundpeak version of this tune, they say “Morgan” instead of “Logan”

Hope this helps!