To nine decimal places, $\displaystyle \log_{10}{2} = 0.301029996$ and $\displaystyle \log_{10}{3} = 0.477121255$

i) Calculate $\displaystyle \log_{10}{5}$ and $\displaystyle \log_{10}{6}$ to three decimal places. By taking logs, or otherwise, show that

$\displaystyle 5 \times 10^{47} < 3^{100} < 6 \times 10^{47}$

Hence write down the first digit of $\displaystyle 3^{100}$

ii) Find the first digit of each of the following numbers: $\displaystyle 2^{1000} $ , $\displaystyle 2^{10 \ 000}$ and $\displaystyle 2^{100 \ 000}$

[OCR STEP I 2000, Question 1]