To solve the equation \( \ln x + \ln x^3 - \ln 4 = 0 \), we can use the properties of logarithms.

1. Combine the logarithmic terms:
   \[
   \ln x + \ln x^3 = \ln(x \cdot x^3) = \ln(x^4)
   \]
   So, the equation becomes:
   \[
   \ln(x^4) - \ln(4) = 0
   \]

2. Use the property of logarithms that states \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \):
   \[
   \ln\left(\frac{x^4}{4}\right) = 0
   \]

3. Exponentiate both sides to eliminate the logarithm:
   \[
   \frac{x^4}{4} = 1
   \]

4. Multiply both sides by 4:
   \[
   x^4 = 4
   \]

5. Take the fourth root of both sides:
   \[
   x = \sqrt[4]{4} = \sqrt{2}
   \]

Thus, the solution is:
\[
x = \sqrt{2}
\]

If you need any details or have questions, let me know! Here are some related questions you might consider:

1. What are the properties of logarithms used in this solution?
2. How would the solution change if the equation were \( \ln x - \ln 4 + \ln x^3 = 0 \)?
3. Can you provide examples of logarithmic equations with different bases?
4. What other methods can be used to solve equations involving logarithms?
5. How can you graph the functions involved in this equation for better visualization?

**Tip:** When solving logarithmic equations, always check for the domain of the variables to ensure all logarithms are defined.

Explore the step-by-step solution to solve the logarithmic equation ln(x) + ln(x^3) - ln(4) = 0 using properties of logarithms. This detailed guide is ideal for students in Grades 10-12 and covers key algebraic concepts and logarithmic properties.

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