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Let's solve the given equation step by step:

The problem is:

\[
2 \ln(2) - \ln(x + 2) = 1
\]

We need to solve for \( x \).

### Step 1: Simplify the equation

We start by moving \( \ln(x + 2) \) to the other side:

\[
2 \ln(2) = 1 + \ln(x + 2)
\]

Now subtract 1 from both sides:

\[
2 \ln(2) - 1 = \ln(x + 2)
\]

### Step 2: Use properties of logarithms

We know that:

\[
2 \ln(2) = \ln(2^2) = \ln(4)
\]

Substitute that into the equation:

\[
\ln(4) - 1 = \ln(x + 2)
\]

Now express \( 1 \) as a logarithm. Since \( \ln(e) = 1 \), we can write it as:

\[
\ln(4) - \ln(e) = \ln(x + 2)
\]

### Step 3: Apply logarithm rules

Using the property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \), we get:

\[
\ln\left(\frac{4}{e}\right) = \ln(x + 2)
\]

Since the logarithms are equal, we can equate the arguments:

\[
x + 2 = \frac{4}{e}
\]

### Step 4: Solve for \( x \)

Now solve for \( x \):

\[
x = \frac{4}{e} - 2
\]

### Step 5: Simplify the expression

To simplify \( x = \frac{4}{e} - 2 \), we get a common denominator:

\[
x = \frac{4 - 2e}{e}
\]

Thus, the value of \( x \) is:

\[
x = \frac{4 - 2e}{e}
\]

### Final Answer:
This matches option **(D)**: 

\[
\boxed{ \frac{4 - 2e}{e} }
\]

Let me know if you'd like further details!

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Here are 5 questions related to this problem:

1. What are the key properties of logarithms used in this solution?
2. How do you express a constant as a logarithmic expression?
3. Why can we equate the arguments of logarithms once their forms match?
4. How can the equation \( 2 \ln(2) \) be simplified using logarithmic rules?
5. What does the base of the natural logarithm, \( e \), represent?

**Tip:** When solving logarithmic equations, always look to simplify by using properties like combining logarithms or exponentiation.

This page provides a detailed solution to the logarithmic equation 2ln(2) - ln(x + 2) = 1. It covers key concepts such as properties of logarithms and algebraic manipulation. Suitable for students in Grades 10-12.

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