To find the value of $\log_3 \frac{1}{9}$, follow these steps:

1. Rewrite the logarithmic expression: $\log_3 \frac{1}{9} = x \implies 3^x = \frac{1}{9}.$

2. Express $\frac{1}{9}$ as a power of 3: $\frac{1}{9} = 3^{-2}.$

3. Substitute this back into the equation: $3^x = 3^{-2}.$

4. Since the bases are the same, equate the exponents: $x = -2.$

Final Answer:

$\log_3 \frac{1}{9} = -2.$

Would you like further details or clarification? Here are some related questions to consider:

1. How do logarithmic and exponential forms relate?
2. Can you calculate $\log_3 27$ or $\log_3 \frac{1}{27}$?
3. What are the properties of logarithms when dealing with fractions?
4. How do logarithms with different bases compare?
5. Can you use change of base formula for this calculation?

Tip: Always try to express the number in terms of the base to simplify logarithmic calculations.