elementary number theory - 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.

Answer

This is probably not the answer you are looking for, and wil probably only be appreciated by people of my age ...

I can still remember from school days that $\log_{10} 2 = 0,30102999$ (I always thought it was noteworthy that it is so close to $0.30103$) - people who went to school in the 1950's can probably recall using logs to base 10 for lots of tedious calculations.

You can then do the multiplication by 4242 without a calculator, and get the fractional part ($=x$, say, but you are likely to need a calculator to find out the first digit of $10^x$, unless you have also memorised $\log 2, \log3, \dots, \log 9$ (I can't!)

Edit:

With a bit more digging in the recesses of my memory, I can just recall that $\log 3$ is something like $0.477$, so $\log 9 = 2 \log 3 = 0.954$, so that should do it ...