Evaluate: $\log _{\sqrt{3}} \frac{1}{27}$

Step 1 ：Given the expression to evaluate is \(\log _{\sqrt{3}} \frac{1}{27}\)

Step 2 ：We can use the change of base formula to simplify this expression. The change of base formula is \(\log_b a = \frac{\log_c a}{\log_c b}\), where \(c\) is the new base.

Step 3 ：In this case, we can choose \(c\) to be 3, because both \(\sqrt{3}\) and \(\frac{1}{27}\) are powers of 3. So, we have \(\log_{\sqrt{3}} \frac{1}{27} = \frac{\log_3 \frac{1}{27}}{\log_3 \sqrt{3}}\).

Step 4 ：Calculating the numerator, we find that \(\log_3 \frac{1}{27} = -3.0\)

Step 5 ：Calculating the denominator, we find that \(\log_3 \sqrt{3} = 0.5\)

Step 6 ：Dividing the numerator by the denominator, we get \(-3.0 / 0.5 = -6.0\)

Step 7 ：Therefore, the value of \(\log _{\sqrt{3}} \frac{1}{27}\) is -6.

Step 8 ：Final Answer: \(\boxed{-6}\)