The expression you provided is \( a \ln(x^2) \), where \( a \) is a constant and \( \ln(x^2) \) represents the natural logarithm of \( x^2 \). We can simplify this logarithmic expression using logarithmic properties.

### Step-by-Step Simplification:
Using the logarithmic identity:
\[
\ln(x^2) = 2 \ln(x)
\]
we can rewrite the expression as:
\[
a \ln(x^2) = a \cdot 2 \ln(x) = 2a \ln(x)
\]

Thus, the simplified form of \( a \ln(x^2) \) is:
\[
2a \ln(x)
\]

### Summary:
\[
a \ln(x^2) = 2a \ln(x)
\]

Would you like more details on how this simplification works, or do you have any questions?

Here are 5 related questions to consider:
1. How does the logarithmic rule \( \ln(a^b) = b \ln(a) \) apply to other bases?
2. What is the derivative of \( a \ln(x^2) \)?
3. How would the expression change if \( x \) were negative or complex?
4. What happens if \( x^2 \) is replaced with another power, like \( x^3 \)?
5. Can you expand \( a \ln(x^2) \) in a Taylor series?

**Tip:** Remember, logarithmic properties can make complex expressions simpler, especially when dealing with powers and roots.

Explore how to simplify the logarithmic expression a ln(x^2), where a is a constant. This step-by-step guide demonstrates the application of logarithmic identities and algebraic simplification techniques. Suitable for students in Grades 10-12.

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