Suppose I have a set of probability mass functions $S_1,\dots,S_n$ and I find that $S_j$ has the greatest entropy using the traditional formula for entropy using $H(S) = \sum_{i\in S}-p_i \log_2(p_i)$.
In this case we're using a base of 2. Is it guaranteed that if instead I used any other base greater than 1 that I would find that $S_j$ has the greatest entropy?
Seems like this would be a needed property in information theory, otherwise it seems like it would be hard to argue we're interested in dividing the space into halves instead of thirds or fifths etc...
Note that $\log_a x=\frac{\ln x}{\ln a}$ so that changing the log base just changes the overall result by a constant factor.