Question

Evaluate :

$lo{g}_{3}4.lo{g}_{4}5.lo{g}_{5}6.lo{g}_{6}7.lo{g}_{7}8.lo{g}_{8}9$

Answer 1

Let $a=lo{g}_{3}4.lo{g}_{4}5.lo{g}_{5}6.lo{g}_{6}7.lo{g}_{7}8.lo{g}_{8}9$

$=\frac{\log 4}{\log 3}.\frac{\log 5}{\log 4}.\frac{\log 6}{\log 5}.\frac{\log 7}{\log 6}.\frac{\log 8}{\log 7}.\frac{\log 9}{\log 8}\dots ({\log }_{b}a=\frac{\log a}{\log b})$

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

$={\log }_{3}9$

$={\log }_3{3}^2$

$=2{\log }_{3}3\dots (\because n{\log }_{a}b={\log }_a{b}^n)$

$a=2$

$\therefore lo{g}_{3}4.lo{g}_{4}5.lo{g}_{5}6.lo{g}_{6}7.lo{g}_{7}8.lo{g}_{8}9=2$