Here are the correct matches for the given items:
- log roads - corduroy roads
- toll roads - turnpikes
- first highway - Lancaster-Philadelphia Turnpike
- ran from Cumberland to Vandalia - Wilderness Road
- ran from Kingsport to Kentucky River - Nicholas Roosevelt
- built the steamboat Clermont - Robert Fulton
- built the steamboat New Orleans - (This one does not match with any in the list; the builder of the New Orleans steamboat is also usually credited to Robert Fulton. If it's referring to a different builder, please clarify.)
Thus, the corrected matches are:
- log roads - corduroy roads
- toll roads - turnpikes
- first highway - Lancaster-Philadelphia Turnpike
- ran from Cumberland to Vandalia - Wilderness Road
- ran from Kingsport to Kentucky River - Nicholas Roosevelt
- built the steamboat Clermont - Robert Fulton
- built the steamboat New Orleans - (best matched with a similar reference to Robert Fulton)