Question

If $x={\log }_{5}7,y={\log }_{7}27,z={\log }_{3}5,showthatxyz=3.$

Answer 1

$xyz={\log }_{5}7{\log }_{7}27{\log }_{3}5$

$xyz=\frac{\log 7}{\log 5}\times \frac{\log 27}{\log 7}\times \frac{\log 5}{\log 3}$ using ${\log }_{a}b=\frac{\log b}{\log a}$

$xyz=\frac{\log 27}{\log 3}$

$xyz=\frac{\log 3^3}{\log 3}$

$xyz=\frac{3\log 3}{\log 3}$ using $\log a^m=m\log a$

$\therefore xyz=3$