Is there a lower and upper bound for the following expression which looks similar to Rayleigh quotient?

Question

The expression looks very similar to that of Rayleigh quotient, but it not the same. Can I find the lower and upper bounds in terms of eigenvalues? If so, how do I proceed?

$ \log \frac{det(B^H S B)}{det(B^H B)} $

Also, note $S$ can be expressed as follows: $ S = y y^H $

Rayleigh quotient: $ \frac{b^H S b}{b^H b} $