To evaluate [tex]\log _{\frac{1}{3}}27[/tex] note that:
- [tex]\dfrac{1}{3}=3^{-1};[/tex]
- [tex]27=3^3.[/tex]
Use following properties:
1) [tex]\log_ab^k=k\log_ab;[/tex]
2) [tex]\log_{a^k}b=\dfrac{1}{k}\log_ab.[/tex]
Then
[tex]\log _{\frac{1}{3}}27=\log_{3^{-1}}3^3=3\log_{3^{-1}}3=3\cdot \dfrac{1}{-1}\log_33=-3\log_33.[/tex]
Since [tex]\log_aa=1,[/tex] you have
[tex]\log _{\frac{1}{3}}27=-3.[/tex]
