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The given inequality is:

\[
\ln(x + 2) - 5 \geq 0
\]

We are tasked with finding the set of solutions on the interval \( ]-2; +\infty[ \).

### Step-by-Step Solution:

1. **Isolate the logarithmic term**:

   \[
   \ln(x + 2) \geq 5
   \]

2. **Exponentiate both sides** to eliminate the logarithm:

   \[
   e^{\ln(x + 2)} \geq e^5
   \]

   Since \(e^{\ln(y)} = y\), this simplifies to:

   \[
   x + 2 \geq e^5
   \]

3. **Solve for \(x\)**:

   \[
   x \geq e^5 - 2
   \]

4. **Consider the domain of the logarithmic function**:
   
   The logarithmic function \( \ln(x + 2) \) is defined only when \(x + 2 > 0\), which implies:

   \[
   x > -2
   \]

5. **Determine the solution set**:

   The inequality \(x \geq e^5 - 2\) gives us the lower bound for \(x\), while the domain \(x > -2\) restricts \(x\) to values greater than \(-2\). However, since \(e^5 - 2 > -2\), the solution is constrained by the lower bound \(x \geq e^5 - 2\).

Thus, the solution set is:

\[
[x_0; +\infty[
\]
where \( x_0 = e^5 - 2 \).

### Conclusion:

The solution set is \( [e^5 - 2; +\infty[ \).

Would you like further details on any step? Here are five related questions you might find interesting:

1. How is the domain of logarithmic functions determined?
2. What is the significance of exponentiation when solving logarithmic inequalities?
3. How does the solution set change for different logarithmic bases?
4. What is the approximate numerical value of \(e^5 - 2\)?
5. Can the logarithmic function have negative arguments?

**Tip:** When solving inequalities involving logarithms, always check both the domain and the result from solving the inequality to ensure that the solution is valid in both cases.

Déterminer l'ensemble des solutions sur ]-2; +∞[ de : ln(x + 2) - 5 ≥ 0

Learn how to solve the logarithmic inequality ln(x + 2) - 5 ≥ 0. Follow a detailed step-by-step solution involving logarithmic properties and exponentiation. Suitable for high school students.

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