The given expression is \( 2\ln(x + 2) - \ln(x + 2) - \ln(2) - 3 \).

Let's simplify it step by step:

### Step 1: Combine Like Terms
We have \( 2\ln(x + 2) \) and \( -\ln(x + 2) \). These terms can be combined:
\[
2\ln(x + 2) - \ln(x + 2) = \ln(x + 2)
\]
So the expression becomes:
\[
\ln(x + 2) - \ln(2) - 3
\]

### Step 2: Use Logarithm Properties
We know the logarithm property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). Applying this to the first two terms:
\[
\ln(x + 2) - \ln(2) = \ln\left(\frac{x + 2}{2}\right)
\]
So the expression now becomes:
\[
\ln\left(\frac{x + 2}{2}\right) - 3
\]

### Final Simplified Expression
The simplified expression is:
\[
\ln\left(\frac{x + 2}{2}\right) - 3
\]

Would you like a detailed explanation or have any further questions?

Here are 5 related questions for practice:
1. How do you simplify \( 3\ln(x) - 2\ln(y) + \ln(2) \)?
2. What is the derivative of \( \ln\left(\frac{x + 2}{2}\right) \)?
3. How do you solve \( 2\ln(x) = 4 \) for \( x \)?
4. What are the properties of logarithms used in simplification?
5. How do you express \( e^{\ln(x) + 3} \) in simpler terms?

**Tip:** When simplifying logarithmic expressions, always look for opportunities to apply logarithmic properties such as the product, quotient, or power rules.

Learn how to simplify the expression 2ln(x + 2) - ln(x + 2) - ln(2) - 3 with detailed step-by-step instructions. This problem involves logarithm properties and is suitable for advanced high school students and above.

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