Q:

Is Big O(logn) log base e?

For binary search tree type of data structures, I see the Big O notation is typically noted as O(logn).  With a lowercase 'l' in log, does this imply log base e (n)  as described by the natural logarithm?  Sorry for the simple question but I've always had trouble distinguishing between the different implied logarithms.

A:

Big O notation is not affected by logarithmic base, because all logarithms in different bases are related by a constant factor, O(ln n) is equivalent to O(log n).

A:

Once expressed in big-O() notation, both are correct.  However, during the derivation of the O() polynomial, in the case of binary search, only log2 is correct.  I assume  this distinction was the intuitive inspiration for your question to begin with.
Also, as a matter of my opinion, writing O(log2 N) is better for your example, because it better communicates the derivation of the algorithm's run-time.
In big-O() notation, constant factors are removed.  Converting from one logarithm base to another involves multiplying by a constant factor.
So O(log N) is equivalent to O(log2 N) due to a constant factor.
However, if you can easily typeset log2 N in your answer, doing so is more pedagogical. In the case of binary tree searching, you are correct that log2 N is introduced during the derivation of the big-O() runtime.  
Before expressing the result as big-O() notation, the difference is very important.  When deriving the polynomial to be communicated via big-O notation, it would be incorrect for this example to use a logarithm other than log2 N, prior to applying the O()-notation. As soon as the polynomial is used to communicate a worst-case runtime via big-O() notation, it doesn't matter what logarithm is used.

A:

It doesn't really matter what base it is, since big-O notation is usually written showing only the asymptotically highest order of n, so constant coefficients will drop away.  Since a different logarithm base is equivalent to a constant coefficient, it is superfluous.
That said, I would probably assume log base 2.

