Question

Evaluate:${\log }_{\frac{1}{27}}\frac{1}{59049‬}$

Answer 1

Let $a={\log }_{\frac{1}{27}}\frac{1}{59049‬}$

$={\log }_{\frac{1}{3^3}}\frac{1}{3^{10}}$

$={\log }_{2^{-3}}{2}^{-10}$

We know that ${\log }_{a^n}{a}^m=\frac{m}{n}{\log }_{a}a$

a$=\frac{-10}{-3}{\log }_{10}10$

since ${\log }_{a}a=1$

a$=\frac{10}{3}$

$\therefore {\log }_{\frac{1}{27}}\frac{1}{59049‬}=\frac{10}{3}$