Question

${\log }_{e}2.{\log }_{x}625={\log }_{10}16.{\log }_{e}10$, then find the value of $x$

Answer 1

Given,

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

or, ${\log }_{e}2.{\log }_x{5}^4={\log }_{e}16$ [ Using the law of logarithm]

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

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

Comparing both sides we get,

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

or, $x^4=5^4$

or, $x=5$.