While trying to analyze an algorithm, I got the following recurrence relation:
$x_{n+1} = x_n - \log_b (x_n)$ and $x_0 = N$ (large), i.e. in every iteration the problem size decreases by a logarithmic part. Now I want to know how many iterations (asymptotically) I will have to perform to *solve* an input of size $N$ (assuming, that I need only a constant number of iterations as soon as $x_i$ becomes $b$), like $argmin_i (x_i < b)$.

I think I was able to show that this behaves as $N / \log_B(N)$ but I was confused that I could find no reference to this problem. Is is too trivial to mention, or did I just not find the right places/names to search for this kind of function/decay?