Question

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

Answer 1

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

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

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

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

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

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