To solve the expression $y = \ln \left( e^{3x \left( \frac{x - 2}{x + 2} \right)^{2/3}} \right)$ and simplify it, let's use logarithmic properties. Here are the steps:

Step 1: Use the Property of Logarithms

The property $\ln(e^a) = a$ can be applied directly to simplify $y$: $y = 3x \left( \frac{x - 2}{x + 2} \right)^{2/3}$

Step 2: Final Answer

After simplification, the function becomes: $y = 3x \left( \frac{x - 2}{x + 2} \right)^{2/3}$

There are no more logarithmic terms in the expression, and this is as simplified as it gets.

Would you like further details on each step or have any questions?

Related Questions

1. What are the properties of logarithms and how are they applied in simplifying expressions?
2. How can you evaluate logarithmic functions involving complex exponents?
3. What are common applications of logarithmic simplifications in calculus?
4. How does the derivative of this function look after simplification?
5. How would the answer change if we had a different base for the logarithm?

Tip: When simplifying logarithmic expressions with exponents, always check if the exponent can be pulled out of the logarithm to simplify the expression faster.