Can we cut off everything as well?

$\begin{aligned} \prod_{r=10}^{99} \log_r (r+1) & = \log_{10} (11) \dot{} \log_{11} (12) \dot{} \log_{12} (13) \dot{} ... \dot{} \log_{99} (100) \\ & = \frac{\log_{10} (11)}{\log_{10} (10)} \dot{} \frac{\log_{10} (12)}{\log_{10} (11)} \dot{} \frac{\log_{10} (13)}{\log_{10} (12)} \dot{} ... \dot{} \frac{\log_{10} (100)}{\log_{10} (99)} \\ & = \frac{\log_{10} (100)}{\log_{10} (10)} \\ & = \boxed{2} \end{aligned}$