Question

If $lo{g}_{3}27.lo{g}_{x}7=lo{g}_{27}x.lo{g}_{7}3,$ then the least value of x, is

Answer 1

$lo{g}_{3}27.lo{g}_{x}7=lo{g}_{27}x.lo{g}_{7}3\Rightarrow \frac{{\log }_{e}27}{{\log }_{e}3}.\frac{{\log }_{e}7}{{\log }_{e}x}=\frac{{\log }_{e}x}{{\log }_{e}27}\frac{{\log }_{e}3}{{\log }_{e}7}\Rightarrow {\log }_{e}x=\frac{{\log }_{e}27}{{\log }_{e}3}{\log }_{e}7\Rightarrow {\log }_{e}x=\frac{3{\log }_{e}3}{3}{\log }_{e}7={\log }_{e}3\ast {\log }_{e}7$