2 line explanation of where e comes from here: to represent a number x in base b, you need roughly log_b(x) digits. If you weight that by the number of different digits in base b, you get b * log_b(x) = b * log(x)/log(b) = b/log(b) * log(x) (where now the log is any base you care to choose).
So assuming x > 1, this is minimized precisely when b/log(b) is minimized. The derivative of b/log(b) is (log(b) - log(e))/log(b)^2, so this is zero when b = e. (The second derivative is 1/(e log(e)) > 0, so this is a minimum).
So assuming x > 1, this is minimized precisely when b/log(b) is minimized. The derivative of b/log(b) is (log(b) - log(e))/log(b)^2, so this is zero when b = e. (The second derivative is 1/(e log(e)) > 0, so this is a minimum).