Question

Solve for $x$:

${\log }_{e}2.{\log }_{x}625={\log }_{10}16.{\log }_{e}10$.

Answer 1

Now,

${\log }_{e}2.{\log }_{x}625={\log }_{10}16.{\log }_{e}10$

or, ${\log }_{e}2.{\log }_x{5}^4={\log }_e{2}^4$

or, ${\log }_{e}2.{\log }_x{5}^4=4{\log }_{e}2$

or, ${\log }_x{5}^4=4$

or, $x^4=5^4$

or, $x=5$.