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!

Learn how to solve the logarithmic equation ln2 + ln(x+2) = -3 with a step-by-step solution. This problem covers logarithmic properties, exponentiation, and algebraic manipulation. Suitable for students in Grades 9-12.

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