Question

If ${\log }_{10}5=0.6990,$ without using log tables, find ${\log }_{10}25,{\log }_{10}4\sqrt{625},{\log }_{10}\left(\frac{1}{3125}\right)$,$\log 5000$ and ${\log }_{10}2$.

Answer 1

${\log }_{10}25={\log }_{10}{5}^2=2{\log }_{10}5=2\times 0.6990=1.3980$

${\log }_{10}\sqrt[4]{4625}={\log }_{10}4\times 25={\log }_{10}100={\log }_{10}{10}^2=2{\log }_{10}10=2$

${\log }_{10}\frac{1}{3125}={\log }_{10}{5}^{-5}=-5{\log }_{10}5=-5\times 0.6990=-3.495$

${\log }_{10}5000={\log }_{10}5\times {10}^3={\log }_{10}5+{\log }_{10}{10}^3=0.6990+3{\log }_{10}10=0.6990+3=3.6990$

${\log }_{10}2={\log }_{10}\frac{10}{5}={\log }_{10}10-{\log }_{10}5=1-0.6990=0.3010$