Question

Show that ${\log }_{b^3}a\times {\log }_{a^3}b\times {\log }_{a^3}c=\frac{1}{27}$.

Answer 1

Now,

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

$=\frac{{\log }_{e}a}{{\log }_e{b}^3}\times \frac{{\log }_{e}b}{{\log }_e{a}^3}\times \frac{{\log }_{e}c}{{\log }_e{a}^3}$ [ Since ${\log }_{x}a=\frac{{\log }_{e}a}{{\log }_{e}x}$]

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

$=\frac{1}{27}$.