It seems to me like questions involving decimal vs binary representations of some number are not particularly interesting: for instance $\pi$ or $\sqrt{2}$ are conjectured to be normal in every base, and as far as I know this is open for any particular base.

On the other hand, in calculating entropy there is again a choice of basis. Further, this gives us certain 'distinguished' real numbers: e.g. the entropy of the Gauss-Kuzmin distribution is $3.432527514776...$ bits, while it is $2.379246769061...$ nats.

Are the properties of these digit strings of the same number 'similar' in any way, or is one 'nicer' in some sense?