Let's solve the problem step by step. The expression given is:

$2\ln(x + 2) - \ln(x + 2) - \ln(2) - 3$

Step 1: Combine like terms

We have $2\ln(x + 2)$ and $-\ln(x + 2)$. These can be combined:

$2\ln(x + 2) - \ln(x + 2) = \ln(x + 2)$

So the expression simplifies to:

$\ln(x + 2) - \ln(2) - 3$

Step 2: Use the logarithmic property

Now, apply the logarithmic property:

$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$

This gives us:

$\ln\left(\frac{x + 2}{2}\right) - 3$

Thus, the simplified expression is:

$\boxed{\ln\left(\frac{x + 2}{2}\right) - 3}$

Final Answer:

The correct option is the first one: $\ln\left(\frac{x+2}{2}\right) - 3$.

Would you like further details, or do you have any questions?

Here are some related questions:

1. What are the main properties of logarithms?
2. How do you simplify expressions involving logarithms and subtraction?
3. How do logarithmic functions behave for different values of $x$?
4. Can this type of expression be solved differently using a calculator?
5. What is the derivative of a logarithmic expression like $\ln(x+2)$?

Tip: Always look for opportunities to combine like terms when simplifying logarithmic expressions to make calculations easier!