Question

Find:
${log}_{25}\sqrt{5}\times {log}_3\frac{1}{5}$

Answer 1

${\log }_{25}\sqrt{5}\times {\log }_5\left(\frac{1}{5}\right)$

$={\log }_{25}\sqrt{5}\times {\log }_5\left(5^{-1}\right)$ since $\frac{1}{a^m}=a^m$

$={\log }_{5^2}{5}^{\frac{1}{2}}\times -{\log }_{5}5$ since $\log a^m=m\log a$

$=\frac{\frac{1}{2}}{2}{\log }_{5}5\times -{\log }_{5}5$ since ${\log }_{a^n}{b}^m=\frac{m}{n}{\log }_{a}b$

$=\frac{-1}{4}$ since ${\log }_{a}a=1$