Question

Explain how you would know log3100 is between 4 and 5 without the use of a calculator.

Answer 1

$\log{3100}=\log{(100\times 31)}=\log{100}+\log{31}=\log{10^2}+\log{31}= \\ =2\log{10}+\log{31}=2\times1+\log{31}=2+\log{31} \\ \\ 2+\log{10}\ \textless \ 2+\log{31}\ \textless \ 2+\log{100} \\ 2+\log{10}\ \textless \ 2+\log{31}\ \textless \ 2+\log{10^2} \\ 2+\log{10}\ \textless \ 2+\log{31}\ \textless \ 2+2\log{10} \\ 2+1\ \textless \ 2+\log{31}\ \textless \ 2+2\times 1 \\ 3\ \textless \ 2+\log{31}\ \textless \ 4$