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It seems that you uploaded an image containing a mathematical expression or question. Since I can't directly access the content of the image in text format, I'll need to extract the equation and assist you further.

The image reads: "Find the difference between \( e^{x^2} \cdot \ln(x - 1) \) and \( e^{x^2} \cdot \ln(x^2) \)."

Let me walk you through how to compute the difference between these two expressions.

We are tasked with finding:

\[
\left( e^{x^2} \cdot \ln(x - 1) \right) - \left( e^{x^2} \cdot \ln(x^2) \right)
\]

### Steps:
1. **Factor out the common term**: Both expressions have a common factor of \( e^{x^2} \), so we can factor it out:
   \[
   e^{x^2} \left( \ln(x - 1) - \ln(x^2) \right)
   \]

2. **Simplify the logarithms**: Use the properties of logarithms to combine the two logarithms. Specifically, we use the rule \( \ln(a^b) = b \ln(a) \) to simplify \( \ln(x^2) \) into \( 2 \ln(x) \):
   \[
   e^{x^2} \left( \ln(x - 1) - 2 \ln(x) \right)
   \]

3. **Final simplified form**: The difference is now in its simplest form:
   \[
   e^{x^2} \left( \ln(x - 1) - 2 \ln(x) \right)
   \]

This is the result.

Would you like me to explain any step further or provide additional details?

### Follow-up Questions:
1. How would you approach differentiating the expression \( e^{x^2} \ln(x - 1) \)?
2. How do logarithmic properties simplify complex expressions involving products or powers?
3. Could you simplify \( \ln(x - 1) - 2 \ln(x) \) further using any other logarithmic rules?
4. What role does the chain rule play when differentiating functions involving exponentials like \( e^{x^2} \)?
5. How can factoring help in simplifying algebraic or trigonometric expressions?

### Tip:
When simplifying logarithmic expressions, always check if you can apply rules like the product, quotient, or power rule of logarithms. This can greatly reduce the complexity of the expression.

Find the difference between e^(x^2) * ln(x - 1) and e^(x^2) * ln(x^2).

Learn how to find the difference between the expressions e^(x^2) * ln(x - 1) and e^(x^2) * ln(x^2). This problem involves simplifying exponential and logarithmic expressions using algebraic properties. Suitable for Grades 11-12 or Early College.

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