To solve the equation $\ln x + \ln(3x) = 6$, we can use the properties of logarithms.

1. Combine the logarithms: $\ln x + \ln(3x) = \ln(3x^2)$ Therefore, the equation becomes: $\ln(3x^2) = 6$

2. Exponentiate both sides to eliminate the natural logarithm: $3x^2 = e^6$

3. Solve for $x^2$: $x^2 = \frac{e^6}{3}$

4. Now, take the square root of both sides: $x = \sqrt{\frac{e^6}{3}} = \frac{e^3}{\sqrt{3}}$

5. To simplify to four decimal places, we calculate: $e^3 \approx 20.0855$ Therefore, $\sqrt{3} \approx 1.7321$ Thus, $x \approx \frac{20.0855}{1.7321} \approx 11.5903$

The final answer is: $\boxed{11.5903}$

Do you want further details or have any questions?

Here are some related questions to consider:

1. How do you use properties of logarithms to combine terms?
2. What is the derivative of the function $\ln(3x^2)$?
3. Can you explain the significance of the natural logarithm in real-world applications?
4. How would the solution change if the equation were $\ln x - \ln(3x) = 6$?
5. What are some other equations involving logarithms and how to solve them?

Tip: Always remember to check if your solution is valid within the context of the original equation, especially when dealing with logarithms, as their domains are restricted.