Hi Amanda! Welcome to the wonderful world of log homes (and maintenance)!
I have seen a number of fixes for this situation. I would suggest tearing off the old shingles and getting down to the original boards. Next install plywood or OSB board onto the roof boards. After that, use a structural 2X 6-8-10 or 12, build up the roof and insulate the cavity. This can be done with sheet foam, fiberglass or blown in foam. Always use a vent system above this of 1 to 2 inches.
Next your finish ply or OSB with shingles or what ever is preferred. A local builder is suggested for this so he can speck out the structural requirements. We would be more than happy to give you a quote to do so.
The Log Doctor.