Question

Find the value of the expression $\frac{2}{{\log }_4(2000{)}^6}+\frac{3}{{\log }_6(2000{)}^6}$.

Answer 1

${\log }_{a}b=\frac{\log b}{\log a}\therefore {\log }_4{\left(2000\right)}^6=\frac{\log {\left(2000\right)}^6}{\log \left(4\right)}{\log }_6{\left(2000\right)}^6=\frac{\log {\left(2000\right)}^6}{\log \left(6\right)}\therefore \frac{2}{\log {\left(2000\right)}^6}+\frac{3}{{\log }_6{\left(2000\right)}^6}=\frac{1}{6\log \left(2000\right)}\left(\log (4^2\times 6^3)\right)=\frac{1}{6}{\log }_{2000}3456=\frac{1}{6}{\log }_{2000}\left(3456\right)$