jmc

Let ${\log }_{3}81=a$. Then $3^a=81=3^4$, so $a=4$. Let ${\log }_3\frac{1}{9}=b$. Then $\frac{1}{9}=3^b$. Express $\frac{1}{9}$ as a power of $3$: $\frac{1}{9}=\frac{1}{3^2}=3^{-2}$. Thus $3^b=3^{-2}$ and $b=-2$. We want to find ${\log }_{3}81-{\log }_3\frac{1}{9}=a-b=(4)-(-2)=\boxed{6}$.