Evaluate the following without using a calculator (show work!): 6. log1/3 81 7. log36 6

$\left(6\right)$ Given expression is ${\log }_{\frac{1}{3}}81$.

We have to evaluate the given expression.

Apply log rule, ${\log }_{\frac{1}{a}}\left(x\right)=-{\log }_a\left(x\right)$

Hence,

$\begin{array}{c}{\log }_{\frac{1}{3}}81=-{\log }_{3}81\\ =-{\log }_3{3}^4\\ =-4{\log }_{3}3\left(\text{Use property }\log m^n=n\log m\right)\\ =-4\times 1\left(\text{Since, }{\log }_{3}3=1\right)\\ =-4\end{array}$

Hence, ${\log }_{\frac{1}{3}}81=-4$.