Solve the equation.
ln x plus ln 3 x equals 6
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Part 1
Select the correct choice and fill in any answer boxes in your choice below. simplify to the fourth decimeals

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.

Solve the equation ln x plus ln 3 x equals 6

Discover how to solve the logarithmic equation ln(x) + ln(3x) = 6 step-by-step. This solution covers logarithmic properties and exponential functions, providing a clear explanation suitable for high school students.

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