 Base dependence of regularity for representations of powers of two

The base-two representations of powers of two form the regular language $10^{\ast }$. Their base-ten representations are not regular. Pumping a fixed decimal substring of length $d$ would produce powers of two $a_i$ satisfying $a_{i+1}-10^d{a}_i=C$. After division by $a_i$ and passage to large $i$, integrality would force a power of two to equal $10^d$, contradicting unique factorization.