Search for The Black Flame mission might be confusing because MHW won't give you a indicator for the location but it's pretty close to where you start it.
1. Get out of the camp using the elevator.
2. Jump down in your left.
3. Jump down again to your right this time and look for the Half-Digested Food in middle of the place.
4. Once you inspect the Half-Digested Food a green trail will lead you to the black flame.
- The Black Flame is underground so this map won't help you much.