For a random number m, you'd expect about 1/10 digits to be zero. Why is it surprising that 2^n would behave somewhat similarly?
you'd have to prove that 2^n has a uniform distribution of digits in base 10.
The numbers 2^n aren't random. For example, the final digit is never 0. You might expect other patterns to appear on larger scales.
The fact that it's linear actually hints that the distribution of digits is actually random: the probability of 0 in any position other than the first and the last is indeed approaching 1/10, as n goes to infinity.