Question

Prove that ${\log }_{b^2}a\times {\log }_{c^3}b\times {\log }_{a^4}c=\frac{1}{24}$

Answer 1

${\log }_{b^2}a\times {\log }_{e^3}b\times {\log }_{a^4}c$

$=\frac{1}{2}{\log }_{b}a\times \frac{1}{3}{\log }_{e}b\times \frac{1}{4}{\log }_{a}c$

$=(\frac{1}{2}\times \frac{1}{3}\times \frac{1}{4})\times (\frac{\log a}{\log b}\times \frac{\log b}{\log c}\times \frac{\log c}{\log a})$

$=\frac{1}{24}$.