We are given the logarithmic equation:

$\log_x \left( \frac{1}{3} \right) = 2$

To solve for $x$, recall the definition of logarithms: if $\log_b(a) = c$, this means $b^c = a$. Applying this to our equation:

$x^2 = \frac{1}{3}$

Now, solve for $x$:

$x = \sqrt{\frac{1}{3}}$

Therefore, $x = \frac{1}{\sqrt{3}}$. To rationalize the denominator:

$x = \frac{\sqrt{3}}{3}$

Thus, the correct answer is:

$\boxed{\frac{\sqrt{3}}{3}}$

However, this exact form isn't listed among the options, but since none of the choices match exactly, it suggests a minor difference in format or approximation. The most fitting answer seems to be the option involving $\sqrt{\frac{1}{3}}$, which aligns with this value after rationalization.

Would you like further details, or do you have any questions?

Related Questions:

1. How do you solve logarithmic equations where the base is unknown?
2. What are the steps for rationalizing the denominator in fractions with square roots?
3. Can you convert logarithmic equations to exponential form in all cases?
4. What are the general properties of logarithms that help in solving such problems?
5. How would this problem change if the logarithmic base was explicitly provided?

Tip:

When solving logarithmic equations, always check if the logarithmic form can be easily converted to an exponential equation for a simpler solution!