Review question

# Can we use $\log_{10}$ to find the first digit of $3^{100}$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9747

## Question

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 $3^{100}$.

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