when discussing average time complexity as in this table.
O (log n)
I'm assuming it is 2 but wanted to verify. Also is it always 2 for these data structures?
I'm in the opinion that the base of the log is not considered a constant so it should be included in the Big O notation.
Whether or not is should be included. What is the actual value?
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.
