The equation is given to be:
[tex]\log _{\frac{1}{3}}6xy[/tex]Recall the logarithm rule:
[tex]\log _{\frac{1}{a}}\mleft(x\mright)=-\log _a\mleft(x\mright)[/tex]Therefore, the expression becomes:
[tex]\log _{\frac{1}{3}}6xy=-\log _36xy[/tex]Factorize the number 6:
[tex]\begin{gathered} 6=2\cdot3 \\ \therefore \\ -\log _36xy=-\log _3(3\cdot2xy) \end{gathered}[/tex]Recall the rule of logarithm:
[tex]\log _c\mleft(ab\mright)=\log _c\mleft(a\mright)+\log _c\mleft(b\mright)[/tex]Thus, the expression becomes:
[tex]-\log _3(3\cdot2xy)=-(\log _33+\log _32+\log _3x+\log _3y)[/tex]Recall the rule:
[tex]\log _aa=1[/tex]Hence, the expression simplifies to give:
[tex]-(\log _33+\log _32+\log _3x+\log _3y)=-(1+\log _32+\log _3x+\log _3y)[/tex]Expanding, we have the answer to be:
[tex]\Rightarrow-1-\log _32-\log _3x-\log _3y[/tex]