In analysis of randomized load balancing allocation (using Chernoff bounds), how do you get upperbound on $\frac{\log n}{\log \log n}$?

Question

This question is from Kelinberg Tardos "Algorithm Design" p. 761. I just don't understand a step they take in their analysis of a randomized load balancing algorithm. I asked my supervisor, who thought it was just a typo. This could be true, but we need something for the bound. I've attached a photo of the relevant page.

I have two questions:

1. How do they get the inequality $\log \log n > 766 \log x$?
2. And on the next line how to get $\frac{\log n}{\log \log n} \leq x$

If it is a typo, how to get upper bound on $\gamma (n)$?

Thank you

Answer 1

I believe the $766$ just shouldn't be there.

Note that for $x>e$ we have $n=x^x > e^x$, implying that $\log n > x$ and $\log \log n > \log x$.

Taking the equation $x \log x = \log n$ and dividing the left side by $\log x$ and the right by $\log \log n$ then gives $$x > \frac{\log n}{\log \log n}.$$