In a Binary Search Tree what is the base of the log in this Big O notation
It doesn't matter what the base of the logarithm is.  $\log_a x = (\log_bx)/\log_b a$ so changing from one (constant) base to another just introduces a constant factor. Big-$O$ and related concepts don't care about constant factors.

The base of the log is absolutely a constant: the base used for the calculations might depend on the whim (or, more likely, convenience) of the person who did the analysis but it certainly doesn't depend on the input.