Are $\log_{10}(x)$ and $\log_2(x)$ in the same big-O class of functions? Are $\log_{10}(x)$ and $\log_{2}(x)$ in the same big-O class of functions? In other words, can one say that $\log_{10}(x)=O(\log x)$ and $\log_{2}(x)=O(\log x)$?
Yes. Because they differ only by a constant factor. Remember high school math: 

$\log_2 x = \dfrac{\log_{10} x}{\log_{10}2}$.