We are given the equation:

$2 \log_3{x} - \log_3{(x+4)} = 2$

Step 1: Apply the properties of logarithms

We can use the power rule of logarithms to simplify the first term:

$2 \log_3{x} = \log_3{x^2}$

Thus, the equation becomes:

$\log_3{x^2} - \log_3{(x+4)} = 2$

Step 2: Apply the quotient rule of logarithms

The quotient rule of logarithms states that:

$\log_b{A} - \log_b{B} = \log_b{\left(\frac{A}{B}\right)}$

Applying this to the equation, we get:

$\log_3\left(\frac{x^2}{x+4}\right) = 2$

Step 3: Convert the logarithmic equation to an exponential form

Using the definition of logarithms, $\log_b{A} = C$ implies $A = b^C$, we convert the equation to:

$\frac{x^2}{x+4} = 3^2$

This simplifies to:

$\frac{x^2}{x+4} = 9$

Step 4: Solve the resulting quadratic equation

Multiply both sides by $x + 4$ to eliminate the fraction:

$x^2 = 9(x + 4)$

Expand the right-hand side:

$x^2 = 9x + 36$

Now, bring all terms to one side:

$x^2 - 9x - 36 = 0$

Step 5: Solve the quadratic equation

We can solve this using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -9$, and $c = -36$.

$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-36)}}{2(1)}$ $x = \frac{9 \pm \sqrt{81 + 144}}{2}$ $x = \frac{9 \pm \sqrt{225}}{2}$ $x = \frac{9 \pm 15}{2}$

Thus, we have two solutions:

$x = \frac{9 + 15}{2} = 12 \quad \text{or} \quad x = \frac{9 - 15}{2} = -3$

Step 6: Check for extraneous solutions

Logarithms are only defined for positive arguments, so $x + 4 > 0$ implies $x > -4$. However, $x = -3$ would make $\log_3{(-3)}$ undefined, so we discard $x = -3$.

Thus, the only valid solution is:

$\boxed{x = 12}$

Do you have any questions or would you like more details?

Here are 5 related questions:

1. How do you solve logarithmic equations using the product and quotient rules?
2. What happens when a logarithmic equation has no real solution?
3. How do you determine the domain restrictions in logarithmic functions?
4. Can you explain how to solve a system involving exponential and logarithmic equations?
5. How does the quadratic formula work, and when is it used?

Tip: Always check if your solution is valid by substituting back into the original logarithmic equation, especially when dealing with logarithms.