Can we use $\log_{10}$ to find the first digit of $3^{100}$?

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To nine decimal places, \(\log_{10}2=0.301029996\) and \(\log_{10}3=0.477121255\).

1. Calculate \(\log_{10}5\) and \(\log_{10}6\) to three decimal places. By taking logs, or otherwise, show that \[5\times 10^{47}<3^{100}<6\times 10^{47}.\]

Hence write down the first digit of \(3^{100}\).

2. Find the first digit of each of the following numbers: \(2^{1000}\); \(2^{10\,000}\); and \(2^{100\, 000}\).