To evaluate [tex]\(\log_3 \frac{1}{27}\)[/tex], we start by expressing [tex]\(\frac{1}{27}\)[/tex] as a power of 3.
First, recall that:
[tex]\[
27 = 3^3
\][/tex]
Thus, the reciprocal of 27 is:
[tex]\[
\frac{1}{27} = \frac{1}{3^3} = 3^{-3}
\][/tex]
Now, we substitute [tex]\(3^{-3}\)[/tex] into the logarithm:
[tex]\[
\log_3 \frac{1}{27} = \log_3 3^{-3}
\][/tex]
We can use the property of logarithms that states [tex]\(\log_b (b^x) = x\)[/tex]. Applying this property here, we get:
[tex]\[
\log_3 3^{-3} = -3
\][/tex]
Therefore, the value of [tex]\(\log_3 \frac{1}{27}\)[/tex] is:
[tex]\[
-3
\][/tex]
The correct answer is:
[tex]\[
\boxed{-3}
\][/tex]
