Here you go
LINK
I have ridden this route, and it's pretty hard to beat. This takes you via US 14 to the Badlands. That is just a short loop, but worth seeing. I find it irresistible not to go too fast along the badlands road to appreciate the scenery. The route goes on the better Black Hills roads to Mt Rushmore and exits via US 16A which you have to see to believe. The road not only switches back, it actually spirals. Before exiting the Black Hills I took you up Hwy 87 the Pinnacles road which has great rock formations and is very technical. You may want to exit
north to see Devils Tower, or use the US 16 exit shown on the map past Jewel Cave.
Rather than directly to Logan, I recommend a northerly route through Gillette and Sheridan to take US14A across the Bighorn Mountains to Red Lodge Montana. This sets you up for the epic Beartooth Pass Highway 212 to Yellowstone's Northeast entry. From there it's straight south on 89 through the Tetons to Logan.
Good luck, let me know if you have any questions. I might have some lodging tips along the way.
Thanks this is very helpful! Lodging tips would be appreciated. In the past I have just thrown a tent on the back, but this trip I plan on treating myself.
Bill