We are given the logarithmic equation:

$\log_x 27 = \frac{3}{4}$

This means that $x$ is the base of the logarithm, and the equation tells us that $x$ raised to the power of $\frac{3}{4}$ equals 27. This can be rewritten as:

$x^{\frac{3}{4}} = 27$

To solve for $x$, we need to eliminate the exponent $\frac{3}{4}$. We can do this by raising both sides of the equation to the power of $\frac{4}{3}$:

$\left( x^{\frac{3}{4}} \right)^{\frac{4}{3}} = 27^{\frac{4}{3}}$

Simplifying both sides:

$x = 27^{\frac{4}{3}}$

Now, we need to compute $27^{\frac{4}{3}}$. We can rewrite 27 as $3^3$:

$x = \left( 3^3 \right)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} = 3^4$

Therefore:

$x = 3^4 = 81$

So, the positive solution for $x$ is:

$\boxed{81}$

Would you like a more detailed breakdown of any step, or do you have any other questions related to logarithms? Here are some related questions to help expand your understanding:

1. What does the expression $\log_b a = c$ mean in general?
2. How can we apply the change of base formula to simplify logarithmic expressions?
3. What are the steps for solving logarithmic equations with fractional exponents?
4. How do we express logarithms in exponential form, and why is it useful?
5. What is the inverse relationship between logarithmic and exponential functions?

Tip: When solving logarithmic equations, always remember to rewrite the logarithmic equation in its exponential form to make the manipulation easier.