Number of rounds of iterative one-round distributed color reduction
A friend helped me, the equation I wrote was correct. I simply didn't develop the result enough to bring it to the desired form. I will post more details when I have time.
Edit: what I needed to achieve the desired result is:

$log(a/b)/log(c/d) = log(b/a)/log(d/c)$

And that we should use Taylor expansion to say $log(1+1/d+1) =~ d+1$ as 1/x+1 -> 0.

If anybody needs help contact me.