Question

Find the value of ${\log }_{3}4.{\log }_{4}5.{\log }_{5}6.{\log }_{6}7.{\log }_{7}8.{\log }_{8}9$.

Answer 1

To find ${\log }_{3}4.{\log }_{4}5.{\log }_{5}6.{\log }_{6}7.{\log }_{7}8.{\log }_{8}9$

Let $L={\log }_{3}4.{\log }_{4}5.{\log }_{5}6.{\log }_{6}7.{\log }_{7}8.{\log }_{8}9$

$=\frac{\log 4}{\log 3}\times \frac{\log 5}{\log 4}\times \frac{\log 6}{\log 5}\times \frac{\log 7}{\log 6}\times \frac{\log 8}{\log 7}\times \frac{\log 9}{\log 8}$ .....$[\because {\log }_{b}a=\frac{\log a}{\log b}]$

$=\frac{\log 9}{\log 3}=\frac{\log 3^2}{\log 3}$

$L=\frac{2\log 3}{\log 3}=2$ .....$[\because \log a^m=m\log a]$

${\log }_{3}4.{\log }_{4}5.{\log }_{5}6.{\log }_{6}7.{\log }_{7}8.{\log }_{8}9$ $=2$