Question

Find the value of the following.
(i) ${\log }_3\left(\frac{1}{81}\right)$
(ii) ${\log }_{7}343$
(iii) ${\log }_6{6}^5$
(iv) ${\log }_{\frac{1}{2}}8$
(v) ${\log }_{10}0.0001$
(vi) ${\log }_{\sqrt{3}}9\sqrt{3}$

Answer 1

(i) ${\log }_3\left(3^{-4}\right)=-4{\log }_{3}3=-4$

(ii) ${\log }_7{7}^3=3{\log }_{7}7=3$

(iii) ${\log }_6{6}^5=5{\log }_{6}6=5$

(iv) ${\log }_{1/2}8={\log }_{2^{-1}}{2}^3=\frac{3}{-1}{\log }_{2}2=-3$

(v) ${\log }_{10}{10}^{-4}=-4{\log }_{10}10=-4$

(vi) ${\log }_{\sqrt{3}}9\sqrt{3}={\log }_{\sqrt{3}}(\sqrt{3}{)}^3=5{\log }_{\sqrt{3}}\sqrt{3}=5$