To solve the equation \( \ln(x^2) + \ln(x) = 0 \), we can first simplify it using logarithmic properties.

### Step-by-step solution:

1. **Combine the logarithmic terms**:
   Recall that \( \ln(a) + \ln(b) = \ln(ab) \). Thus,
   \[
   \ln(x^2) + \ln(x) = \ln(x^2 \cdot x) = \ln(x^3)
   \]
   The equation now becomes:
   \[
   \ln(x^3) = 0
   \]

2. **Exponentiate both sides**:
   To remove the logarithm, exponentiate both sides of the equation:
   \[
   e^{\ln(x^3)} = e^0
   \]
   This simplifies to:
   \[
   x^3 = 1
   \]

3. **Solve for \(x\)**:
   To solve for \(x\), take the cube root of both sides:
   \[
   x = 1
   \]

Thus, the solution to the equation is \( x = 1 \).

---

Do you want more details on any of the steps or further clarification?

Here are some related questions:
1. What is the derivative of \( \ln(x^2) + \ln(x) \)?
2. How does the property \( \ln(a) + \ln(b) = \ln(ab) \) work?
3. What happens if \( \ln(x^2) + \ln(x) = \ln(c) \) instead of 0?
4. Can you solve \( \ln(x^n) + \ln(x) = 0 \) for any value of \( n \)?
5. What is the general rule for solving logarithmic equations?

**Tip**: Remember, \( \ln(x) \) is only defined for \( x > 0 \), so when solving logarithmic equations, check that your solutions are in the domain.

Learn how to solve the logarithmic equation ln(x^2) + ln(x) = 0 with a clear, step-by-step explanation. This problem highlights key logarithmic properties and shows how to simplify and solve using exponentiation. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation