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!

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.