Question

Given ${\log }_{10}2={}^{.}30103,$ find ${\log }_{10}7={}^{.}84509,$ find ${\log }_7\sqrt{2}$ and ${\log }_{\sqrt{2}}7.$

Answer 1

${\log }_7\sqrt{2}=\frac{\log \sqrt{2}}{\log 7}=\frac{\frac{1}{2}\log 2}{\log 7}=\frac{\frac{1}{2}\times 0.30103}{0.84509}=0.1781$

${\log }_{\sqrt{2}}7=\frac{1}{{\log }_7\sqrt{2}}=\frac{1}{0.1781}=5.615$