The number N = 6 log₁₀2 + log₁₀31, lies between two successive integers whose sum is equal to

A

5

B

7

C

9

[ CORRECT ANS IS (B) ]

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Approximate} \mathrm{values} \mathrm{of}: \\ {\log }_{10}2=0.30 {\log }_{10}3=0.48 \\ \mathrm{Also} \\ {\log }_{10}10=1 \left[\mathrm{This} \mathrm{is} \mathrm{exact} \mathrm{value}\right] \\ \mathrm{N}=6{\log }_{10}2+{\log }_{10}\left(31\right) \\ =6{\log }_{10}2+{\log }_{10}\left(3.1\times 10\right) \\ =6{\log }_{10}2+{\log }_{10}\left(3.1\right)+{\log }_{10}10 \left[\log \left(\mathrm{mn}\right)=\log \left(\mathrm{m}\right)+\log \left(\mathrm{n}\right)\right] \\ \mathrm{Writing} \mathrm{approximate} \mathrm{values} \mathrm{we} \mathrm{get}: \\ =6\times 0.30+0.49+1 \\ =1.8+0.49+1 \\ =3.29 \\ \mathrm{which} \mathrm{lies} \mathrm{between} 3 \mathrm{and} 4. \\ \mathrm{Hence} 3+4=7 \\ \Rightarrow \mathbf{Option}\left(\mathbf{B}\right)\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{correct}\mathbf{.} \end{gathered}$$

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