Leading Digit of $2^{4242}$ How could I solve this problem?

Find the first digit of $2^{4242}$ without using a calculator.

I know how to find the last digit with modular arithmetic, but I can't use that here.

If you have memorized that $\log_{10}2\approx 0.30103$ you can multiply by $4242$ and take the fractional part as $0.9692$ which looks like it should be greater than $\log_{10}9$ (and it is, but it is closer than I would have thought, it is $\approx 0.9542$).  I don't know how to do it without log tables.
