We will solve the equation:

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

Step 1: Isolate the $x$-term

Subtract $\ln(2)$ from both sides: $-3x = 2 - \ln(2)$

Step 2: Solve for $x$

Divide both sides by $-3$: $x = \frac{\ln(2) - 2}{3}$

Final Answer:

$x = \frac{\ln(2) - 2}{3}$

If you need a numerical approximation, we can compute the value using $\ln(2) \approx 0.693$. Let me know if you'd like me to proceed with that! 😊

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Questions for Expansion:

1. What does $\ln(x)$ represent, and how is it different from $\log(x)$?
2. How would this solution change if the equation were $\ln(2) + 3x = 2$?
3. What happens if we have a base other than $e$ for the logarithm?
4. How can you verify the solution using substitution?
5. What is the significance of $\ln(2)$ in mathematics?

Tip:

When solving logarithmic equations, always isolate the logarithmic term or the variable for clarity.