Question

If ${\log }_{3}4\cdot {\log }_{4}5\cdot {\log }_{5}6\cdots {\log }_{n-1}\left(n\right)=4$, then the value of $n$ is

Answer 1

${\log }_{3}4\cdot {\log }_{4}5\cdot {\log }_{5}6\cdots {\log }_{n-1}\left(n\right)=4$
$\Rightarrow {\log }_{3}4\cdot \frac{{\log }_{3}5}{{\log }_{3}4}\cdot \frac{{\log }_{3}6}{{\log }_{3}5}\cdots \frac{{\log }_{3}n}{{\log }_3(n-1)}=4$
$\Rightarrow {\log }_{3}n=4$
$\Rightarrow n=3^4=81$