Problem 44714. The Generalized-N Eggs Problem

A firm has invented a super-strong egg. For publicity purposes, it wants to determine the highest floor in a story building with N_f floors from which such an egg can fall without breaking. The firm has given a tester N identical eggs to experiment with. The same egg can be dropped multiple times unless it breaks.

What is the minimum number of droppings d that is guaranteed to determine the highest safe floor in all cases for a given number of floors N_f, and eggs N ?

APPLICATION: This problem has many applications in the real world such as avoiding a call out to the slow HDD, or attempting to minimize cache misses, or running a large number of expensive queries on a database.