To solve the equation $\log_x 27 = -\frac{1}{3}$, follow these steps:

Step 1: Rewrite the logarithmic equation in exponential form

The equation $\log_x 27 = -\frac{1}{3}$ can be rewritten in its exponential form:

$x^{-\frac{1}{3}} = 27$

Step 2: Eliminate the negative exponent

To eliminate the negative exponent, take the reciprocal of both sides:

$x^{\frac{1}{3}} = \frac{1}{27}$

Step 3: Cube both sides to solve for $x$

Now, cube both sides of the equation to solve for $x$:

$x = \left(\frac{1}{27}\right)^3$

Step 4: Simplify the expression

Simplify the expression:

$x = \frac{1}{27^3} = \frac{1}{19683}$

So, the solution is:

$\boxed{x = \frac{1}{19683}}$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you solve logarithmic equations with different bases?
2. What is the general form of converting a logarithmic equation to an exponential equation?
3. How can logarithmic properties be used to simplify complex logarithmic expressions?
4. How do you solve logarithmic equations involving multiple logarithms?
5. What are the rules for exponents that are useful when solving logarithmic equations?
6. How do you solve logarithmic equations graphically?
7. What is the importance of the base in logarithmic functions?
8. How do you deal with logarithmic equations that have fractional exponents?

Tip: Always remember that logarithms can be converted to exponential form, which often simplifies the process of solving equations.