Not sure how to interpret the question: how to find egg-breaking height
In this task, the resources that you care about are not running time, but rather the number of throws and the number of eggs broken (in the worst case). Here is an example. Suppose that $N = 2$. An optimal algorithm, which throws at most two eggs and breaks at most one egg, goes as follows:


Throw an egg from floor 1. If it breaks, the answer is $F=1$. Otherwise, continue.
Throw an egg from floor 2. If it breaks, the answer is $F=2$. Otherwise, the answer is $F=3$.


(If $F=3$ is not a possibility, then we need only one throw.)

In this way, we can determine $F$ using at most $\sim N$ throws and one broken egg. There is a way to improve the number of throws to only $\sim \log N$, at the cost of potentially breaking up to $\sim \log N$ eggs.