We are given the equation:

$\log_4(\log_3{x}) = 1$

Step 1: Convert the logarithmic equation.

First, rewrite the equation using the definition of logarithms. The equation $\log_4(A) = 1$ means that $4^1 = A$, so we can rewrite the equation as:

$\log_3{x} = 4^1 = 4$

Step 2: Solve for $x$.

Now, we have $\log_3{x} = 4$. Using the definition of logarithms again, this means that:

$x = 3^4 = 81$

Thus, the solution to the equation is:

$x = 81$

Let me know if you'd like more details on any step! Here are some related questions to further explore:

1. What is the general formula for solving $\log_b(x) = y$?
2. How does the change of base formula work for logarithms?
3. What are the key properties of logarithms?
4. How can logarithmic equations be applied to real-world problems?
5. Can you solve $\log_2(\log_5{x}) = 2$?

Tip: Always remember to check your answers by plugging them back into the original equation to verify they satisfy it!