Question

Which is greater :
$lo{g}_{4}5orlo{g}_{1/16}\left(1/25\right).$

Answer 1

${\log }_{4}5=\frac{\log 5}{\log 4}=\frac{\log 5}{\log 2^2}=\frac{\log 5}{2\log 2}=\frac{\log \left(\frac{10}{2}\right)}{2\log 2}=\frac{\log 10-\log 2}{2\log 2}=\frac{1-0.3}{2\times 0.3}=\frac{.7}{.6}=\frac{7}{6}{\log }_{\frac{1}{16}}\left(\frac{1}{25}\right)=\frac{\log \left(\frac{1}{5^2}\right)}{\log \left(\frac{1}{2^4}\right)}=\frac{\log \left(5^{-2}\right)}{\log \left(2^{-4}\right)}=\frac{-2\log 5}{-4\log 2}=\frac{-2\log \left(\frac{10}{2}\right)}{-4\times 0.3}=\frac{-2[\log 10-\log 2]}{-1.2}=\frac{-2[1-0.3]}{-1.2}=\frac{-1.4}{-1.2}=\frac{7}{6}$

Both are same.