Most efficient base to compute from...

Assume there are V independent states of information, then we can present V/N digits in base N.

The amount of information we can represent: I=NV/N$I=N^{V/N}$

The value of N that makes I the maximum is the most "efficient" base

Natural log both sides: lnI=(V/N)lnN

Take the derivative: (lnI)'=V(1-lnN)/N2$N^2$

When lnI gets to an extreme, (lnI)'=0, i.e. N=e

Take the second derivative: (lnI)''=V(2lnN-3)/N3$N^3$

When N=e, (lnI)''=−1/N3$-1/N^3$, which is negative

Therefore, I reaches its maximum when N=e.

ALSO SEE: https://www.americanscientist.org/issues/pub/2001/6/third-base/1