# Base of logarithm in runtime of Prim's and Kruskal's algorithms

For Prim's and Kruskal's Algorithm there are many implementations which will give different running times. However suppose our implementation of Prim's algorithm has runtime $O(|E| + |V|\cdot \log(|V|))$ and Kruskals's algorithm has runtime $O(|E|\cdot \log(|V|))$.

What is the base of the $\log$?

• It doesn't matter. When they are inside O-notation like this, all logarithms with constant base are equivalent.
– David Eppstein
Commented Oct 27, 2012 at 21:14
• That said, if you want to know which base the logarithm is before O-ing it away, you'd have to look/give us the specific implementation. Commented Nov 1, 2012 at 22:49
• See also the question Are log10(x) and log2(x) in the same big-O class of functions?
– Juho
Commented Nov 1, 2012 at 23:38

If the log were initially in base $b$, it is asymptotically the same as if it were in base $k$:
$$O(\log_b n) = O\left(\frac{\log_k n}{\log_k b}\right) = O(\log_k n)$$
because $k$ and $b$ are both constants; $\log_k b = O(1)$.