We are given the equation:

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

To solve for $x$, let's follow these steps:

Step 1: Use the logarithmic property

Use the property of logarithms that says $\ln(a) + \ln(b) = \ln(a \cdot b)$.

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

So the equation becomes:

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

Step 2: Exponentiate both sides

To eliminate the natural logarithm, exponentiate both sides of the equation using base $e$:

$2(x + 2) = e^{-3}$

Step 3: Solve for $x$

Now, simplify and solve for $x$:

$2(x + 2) = \frac{1}{e^3}$

Divide both sides by 2:

$x + 2 = \frac{1}{2e^3}$

Now, subtract 2 from both sides:

$x = \frac{1}{2e^3} - 2$

Final Answer:

$x = \frac{1}{2e^3} - 2$

If you need further simplification, you can use the approximate value of $e^3 \approx 20.0855$, but the exact form is often preferred in mathematical solutions.

Would you like more details or further clarifications?

Here are some related questions to deepen your understanding:

1. What is the value of $\ln(e^x)$, and how does it simplify?
2. How do you solve logarithmic equations involving products or quotients?
3. Can you solve $\ln(a) + \ln(b) = c$ for other base $c$ values?
4. How does exponentiation affect natural logarithms?
5. What real-world problems involve solving logarithmic equations?

Tip: Always remember the fundamental properties of logarithms to simplify equations efficiently!