I was wondering if for almost all exponents $x$, $2^x$ written in decimal would contain at least one 0 digit.
I've checked with a computer up to $ 10^5 $ and I didn't find any exponent which doesn't contain a 0 greater than 86.
Is that correct generally? How can you prove/disprove it?
I intuitively think it's correct, because as a number has more base 10 digits I would expect it to be more rare for none of its digit to be 0.