We are tasked with solving the logarithmic equation:

\[
\log_4{(x+6)} + \log_4{(x)} = 2
\]

### Step 1: Use the logarithmic product rule
The logarithmic product rule states that:

\[
\log_b{(a)} + \log_b{(c)} = \log_b{(a \cdot c)}
\]

Applying this rule to our equation:

\[
\log_4{[(x+6) \cdot x]} = 2
\]

Simplifying the expression inside the logarithm:

\[
\log_4{(x^2 + 6x)} = 2
\]

### Step 2: Rewrite the logarithmic equation in exponential form
The logarithmic equation \(\log_b{(a)} = c\) is equivalent to \(b^c = a\). Applying this property:

\[
4^2 = x^2 + 6x
\]

This simplifies to:

\[
16 = x^2 + 6x
\]

### Step 3: Rearrange into a standard quadratic equation
Move all terms to one side of the equation:

\[
x^2 + 6x - 16 = 0
\]

### Step 4: Solve the quadratic equation
We now solve \(x^2 + 6x - 16 = 0\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \(a = 1\), \(b = 6\), and \(c = -16\), we have:

\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-16)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 + 64}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-6 \pm 10}{2}
\]

Thus, the two possible solutions are:

\[
x = \frac{-6 + 10}{2} = \frac{4}{2} = 2
\]
\[
x = \frac{-6 - 10}{2} = \frac{-16}{2} = -8
\]

### Step 5: Check for extraneous solutions
Logarithms are only defined for positive arguments, so we need to check the solutions:

- For \(x = 2\), both \(\log_4{(2+6)}\) and \(\log_4{(2)}\) are valid.
- For \(x = -8\), \(\log_4{(-8)}\) is not defined, so this solution is extraneous.

### Final Solution:
The only valid solution is:

\[
x = 2
\]

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

### Related Questions:
1. How do you graph logarithmic functions?
2. What are the other logarithmic rules besides the product rule?
3. Can we solve logarithmic equations with different bases?
4. How do you simplify logarithmic expressions involving radicals?
5. What are the conditions for extraneous solutions in logarithmic equations?

### Tip:
When solving logarithmic equations, always check if the arguments of the logarithms are positive, as logarithms are not defined for negative numbers or zero.

Learn how to solve the logarithmic equation log4(x+6) + log4(x) = 2 using logarithmic rules and the quadratic formula. This step-by-step guide explores key concepts like the logarithmic product rule, exponential form, and solving quadratic equations. Ideal for high school students in Grades 10-12.

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