In a ΜAΘ contest from 1991, I found this problem in my problem book. I know how to solve problems like this, and I know how to solve it if the problem tells me to find the digits in $2^{44}$, but $5^{44}$ makes me think about the problem like this:
$${5^{44} = \frac{10^{44}}{2^{44}}}$$
Given this, I don't think I can begin to approximate it with a change of base, since I'm only given ${\log_{10} 2}\approx {0.3010}$, not ${\log_2 10}$ which I could calculate. Maybe I could try $44$ digits from the $10^{44}$ minus the digits in $2^{44}$?
$${44} - 2^{44\log_210}$$
I don't think thats how division works either, I'm very lost. Please do help.
You are on the right track.
Note that we want to compute $$\log_{10}(5^{44}) = 44\log_{10}(5) = 44 \log_{10}(10 / 2) = 44(\log_{10}(10) - \log_{10}(2)) = 44(1 - 0.3010) \approx 30.7$$ so we get 31 digits.