Question

If $\left({\log }_{2}3\right)\left({\log }_{3}4\right)\left({\log }_{4}5\right)...\left({\log }_n(n+1)\right)=10,$ find $n$

Answer 1

$\Rightarrow \frac{\log 2}{\log 3}\times \frac{\log 3}{\log 4}\times \frac{\log 4}{\log 5}\times ...\frac{\log n}{\log n+1}=\frac{1}{10}$

$\Rightarrow \frac{\log 2}{\log 3}\times \frac{\log 3}{\log 4}\times \frac{\log 4}{\log 5}\times ...\frac{\log n-1}{\log n}\frac{\log n}{\log n+1}=\frac{1}{10}$

$\Rightarrow \frac{\log 2}{\log n+1}=\frac{1}{10}$

$\Rightarrow n+1=2^{10}=1024$

$\Rightarrow n=1024-1=1023$

$\therefore n=1023$