Question

If ${\log }_{27}625+7^{{\log }_{11}13}$ and $B={\log }_{9}125+{13}^{{\log }_{11}7},$ then find the relation between A and B.

Answer 1

$A={\log }_{27}625+7{\log }_{{11}^{13}}={\log }_{3^3}{5}^4+7^{{\log }_{11}13}$

$\Rightarrow A=\frac{4}{3}{\log }_{3}5+7^{{\log }^{11}13}$ .....(i)

$B={\log }_{9}125+{13}^{{\log }_{11}7}$

$\Rightarrow B={\log }_{3^2}{5}^3+7{\log }_{{11}^{13}}$

$\Rightarrow B=\frac{3}{2}{\log }_{3}5+7^{{\log }_{11}13}$ .....(ii)

By (i) and (ii) we have,

$A-\frac{4}{3}{\log }_{3}5=B-\frac{3}{2}{\log }_{3}5$

$\because \frac{4}{3}{\log }_{3}5<\frac{3}{2}{\log }_{3}5$

$\therefore A>B.$