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The image contains a series of mathematical expressions, mostly dealing with logarithms. Let's go through each of them to either simplify or solve them, as appropriate:

1. \(\ln(e^5)\)
   - Solution: \(\ln(e^5) = 5 \cdot \ln(e) = 5\)

2. \(\ln\left(\frac{e^7}{e^2}\right)\)
   - Solution: \(\ln\left(\frac{e^7}{e^2}\right) = \ln(e^{7-2}) = \ln(e^5) = 5\)

3. \(\ln\left(\frac{7}{11}\right)\)
   - This is already simplified. It's \(\ln\left(\frac{7}{11}\right)\).

4. \(\ln(4x) + \ln(2x)\)
   - Solution: \(\ln(4x) + \ln(2x) = \ln(4x \cdot 2x) = \ln(8x^2)\)

5. \(\ln\left(\frac{(x+2)^6}{x^2}\right)\)
   - Solution: \(\ln\left(\frac{(x+2)^6}{x^2}\right) = \ln((x+2)^6) - \ln(x^2) = 6\ln(x+2) - 2\ln(x)\)

6. \(\ln\left(\frac{x^3}{(y+2)^4}\right)\)
   - Solution: \(\ln\left(\frac{x^3}{(y+2)^4}\right) = 3\ln(x) - 4\ln(y+2)\)

7. \(2\ln(x^2) - 3\ln(x^3)\)
   - Solution: \(2\ln(x^2) - 3\ln(x^3) = 4\ln(x) - 9\ln(x) = -5\ln(x)\)

8. \(\ln(10x) - 2\ln(5x)\)
   - Solution: \(\ln(10x) - 2\ln(5x) = \ln(10x) - \ln(25x^2) = \ln\left(\frac{10x}{25x^2}\right) = \ln\left(\frac{2}{5x}\right)\)

9. \(\ln\left(\frac{e^{2x}}{e^3}\right)\)
   - Solution: \(\ln\left(\frac{e^{2x}}{e^3}\right) = \ln(e^{2x-3}) = 2x - 3\)

10. \(\ln\left(\frac{x^2}{x^4}\right)\)
    - Solution: \(\ln\left(\frac{x^2}{x^4}\right) = \ln(x^{2-4}) = \ln(x^{-2}) = -2\ln(x)\)

Would you like further details on any specific problem or any additional questions? Here are some related questions you might explore:

1. What is the derivative of the logarithmic function \( \ln(x) \)?
2. How do logarithmic properties simplify complex logarithmic expressions?
3. What is the relationship between logarithms and exponents?
4. How can logarithmic functions be applied to solve real-world problems?
5. What are the differences between natural logarithms and common logarithms?

**Tip:** When simplifying logarithmic expressions, always look for opportunities to apply properties like the product, quotient, and power rules. These can greatly simplify complex expressions.

The image contains various logarithmic expressions, including simplifications and operations involving logarithms.

Learn how to simplify various logarithmic expressions using the properties of logarithms. This problem set covers key concepts such as the product rule, quotient rule, and power rule, making it ideal for students in Grades 10-12.

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