Question

If $3^a=4,4^b=5,5^c=6,6^d=7,7^e=8$ and $8^f=9,$ then the value of the product $\left(abcdef\right)$ is

Answer 1

$3^a=4\Rightarrow a={\log }_{3}4$
Similarly,
$b={\log }_{4}5c={\log }_{5}6d={\log }_{6}7e={\log }_{7}8f={\log }_{8}9$

$\therefore \left(abcdef\right)={\log }_{3}4\cdot {\log }_{4}5\cdot {\log }_{5}6\cdot {\log }_{6}7\cdot {\log }_{7}8\cdot {\log }_{8}9$

$=\frac{\log 4}{\log 3}\cdot \frac{\log 5}{\log 4}\cdot \frac{\log 6}{\log 5}\cdot \frac{\log 7}{\log 6}\cdot \frac{\log 8}{\log 7}\cdot \frac{\log 9}{\log 8}\cdot$

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