IMAS 2023–2024 Junior Division — Round 2 Question 6

It is known that $2^{84}$ is a 26-digit number so that 1 is its first digit. Among $2^1,2^2,2^3,2^4,\ldots,2^{83}$ and $2^{84}$, how many numbers are there so that 4 is their first digit?