To solve the equation \( 3\ln(x) - 2 = 0 \) for \( x \), follow these steps:

1. **Isolate the logarithm**:
   \[
   3\ln(x) = 2
   \]

2. **Divide both sides by 3**:
   \[
   \ln(x) = \frac{2}{3}
   \]

3. **Exponentiate both sides** to get rid of the natural logarithm (since \( \ln(x) \) is the natural logarithm, the base of the exponentiation is \( e \)):
   \[
   x = e^{\frac{2}{3}}
   \]

So, the exact solution is:
\[
x = e^{\frac{2}{3}}
\]

Would you like any details or have any questions about this solution?

Here are five related questions you might find interesting:

1. How would you solve \( 4\ln(x) + 1 = 0 \)?
2. What is the approximate value of \( e^{\frac{2}{3}} \)?
3. Can you solve \( 5\ln(x) = 3\ln(2) \)?
4. How would you solve \( \ln(x^2) = 4 \)?
5. What happens if you take the logarithm of both sides of the equation \( x^3 = 10 \)?

**Tip:** Remember that \( \ln(x) \) is only defined for \( x > 0 \), so when solving logarithmic equations, always check that your solutions are positive.

Learn how to solve the equation 3ln(x) - 2 = 0 step-by-step with a detailed solution. Understand logarithmic equations and the process of isolating logarithms to find the exact solution. Suitable for high school students.

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