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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!

Explore how to simplify logarithmic expressions such as 2ln(x + 2) - ln(x + 2) - ln(2) with a detailed step-by-step solution. This guide covers the use of logarithmic identities and properties to simplify expressions, suitable for high school students in Grades 10-12.

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