Question

Evaluate: ${\log }_{3}2{\log }_{4}3{\log }_{5}4...{\log }_{64}63$

Answer 1

Given, ${\log }_{3}2{\log }_{4}3{\log }_{5}4...{\log }_{64}63$

$=\frac{\log 2}{\log 3}\times \frac{\log 3}{\log 4}\times \frac{\log 4}{\log 5}\times ...\times \frac{\log 63}{\log 64}$

$={\log }_{64}2$

$={\log }_{2^6}2$

$=\frac{1}{6}{\log }_{2}2$

$=\frac{1}{6}$