It seems like you're trying to write an equation involving a natural logarithm (\(\ln\)) and a seventh root of \(x\). Let me clarify it for you:

You probably mean:

\[
\ln\left(\sqrt[7]{x}\right) = ?
\]

This expression involves the natural logarithm of the seventh root of \(x\). Using logarithmic rules, you can simplify it as follows:

\[
\ln\left(\sqrt[7]{x}\right) = \ln(x^{1/7})
\]

Then, apply the logarithmic power rule:

\[
\ln(x^{1/7}) = \frac{1}{7} \ln(x)
\]

So, the simplified expression is:

\[
\ln\left(\sqrt[7]{x}\right) = \frac{1}{7} \ln(x)
\]

Would you like more details or have any questions about this? Here are five related questions to expand your understanding:

1. What are the other logarithmic rules, and how do they apply to various functions?
2. How would you solve an equation like \(\ln\left(\sqrt[7]{x}\right) = 2\)?
3. What is the significance of natural logarithms in calculus?
4. How do logarithmic and exponential functions relate to each other?
5. How do you differentiate functions that involve logarithms like \(\ln(\sqrt[7]{x})\)?

**Tip:** Logarithms are very useful in solving exponential equations and simplifying expressions involving powers!

Learn how to simplify the natural logarithm of the 7th root of x using logarithmic rules. This detailed explanation covers key algebraic concepts like logarithmic power rules and roots. Ideal for high school students in Grades 10-12.

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