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.