To solve the equation \( 5 + \ln(-3x) = 2 \) for \( x \), we follow these steps:

1. Subtract 5 from both sides of the equation:
   \[
   \ln(-3x) = 2 - 5
   \]
   \[
   \ln(-3x) = -3
   \]

2. Exponentiate both sides to remove the natural logarithm:
   \[
   -3x = e^{-3}
   \]

3. Solve for \( x \):
   \[
   x = -\frac{e^{-3}}{3}
   \]
   \[
   x = -\frac{1}{3e^3}
   \]

Thus, the solution is:
\[
x = -\frac{1}{3e^3}
\]

Do you want details or have any questions?

Here are 5 numbered questions you might consider asking next:
1. How do you solve logarithmic equations in general?
2. Can you explain the properties of natural logarithms?
3. How does exponentiation help in solving logarithmic equations?
4. What are some common mistakes when solving logarithmic equations?
5. Can you solve an equation involving both logarithms and exponents?

**Tip:** When solving logarithmic equations, always check if the argument of the logarithm is positive, as the logarithm of a non-positive number is undefined in the real number system.

Learn how to solve the equation 5 + ln(-3x) = 2 for x with a step-by-step solution. This problem involves logarithms, exponentiation, and equations, suitable for advanced high school students.

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