The equation you've provided is:

\[
x \log(4x + 12) = 0
\]

To solve this equation, follow these steps:

### Step 1: Set \( \log(4x + 12) = 0 \)

Since the product of \( x \) and \( \log(4x + 12) \) equals zero, either \( x = 0 \) or \( \log(4x + 12) = 0 \).

### Step 2: Solve for \( x \)

#### Case 1: \( x = 0 \)

Substituting \( x = 0 \) into the original equation:
\[
0 \log(4(0) + 12) = 0
\]
This holds true, so \( x = 0 \) is a valid solution.

#### Case 2: \( \log(4x + 12) = 0 \)

We know that \( \log(a) = 0 \) when \( a = 1 \), so:
\[
4x + 12 = 1
\]
Solving for \( x \):
\[
4x = 1 - 12
\]
\[
4x = -11
\]
\[
x = -\frac{11}{4}
\]

### Step 3: Final Solution

The solutions to the equation are:
\[
x = 0 \quad \text{or} \quad x = -\frac{11}{4}
\]

Would you like further details on any step?

### 5 Related Questions:
1. How do logarithmic properties affect equations?
2. What happens when logarithms yield negative values?
3. How would this equation change if the logarithmic base was different?
4. Can logarithmic equations have more than two solutions?
5. What are the steps for solving non-logarithmic terms inside the log function?

### Tip:
Always remember to check the domain of a logarithmic function—logarithms are only defined for positive arguments.

Learn how to solve the logarithmic equation x log(4x + 12) = 0 step-by-step. Understand the use of the Zero Product Property and logarithmic rules. This detailed solution is suitable for high school students in Grades 10-12, covering key concepts in algebra and logarithmic equations.

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