Without using log tables, prove that 2 5 < _ 10 3<1 2 .

Question

Without using log tables, prove that $\frac{2}{5}<\log _{10} 3<\frac{1}{2}$.

✓

Answer

We have to prove that, $\frac{2}{5}<\log _{10} 3<\frac{1}{2}$
i.e., to prove that $\frac{2}{5}<\log _{10} 3$ and $\log _{10} 3<\frac{1}{2}$
i.e., to prove that $2<5 \log _{10} 3$ and $2 \log _{10} 3<1$
i.e., to prove that $2 \log _{10} 10<5 \log _{10} 3$ and $2 \log _{10} 3<\log _{10} 10 \ldots \ldots .\left[\because \log _a\right.$ a $=1]$
i.e., to prove that $\log _{10} 10^2<\log _{10} 3^5$ and $\log _{10} 3^2<\log _{10} 10$
i.e., to prove that $10^2<3^5$ and $3^2<10$
i.e., to prove that $100<243$ and $9<10$ which is true
$
\therefore \frac{2}{5}<\log _{10} 3<\frac{1}{2}
$

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