https://file.aiquickdraw.com/mathbot/file/1725428994964r0pgo6zj.jpg

The expression in the image is:

\[
\log_{n^2}(n^2) - 4 \log_n\left(\frac{1}{n}\right)
\]

We are asked to find the value of this expression when \( n > 1 \).

Let's break down the expression step by step.

1. **First term**: \(\log_{n^2}(n^2)\)

   The logarithm of a number with the same base is always 1. So,

   \[
   \log_{n^2}(n^2) = 1
   \]

2. **Second term**: \(4 \log_n\left(\frac{1}{n}\right)\)

   The logarithm \(\log_n\left(\frac{1}{n}\right)\) can be rewritten using the property of logarithms:
   
   \[
   \log_n\left(\frac{1}{n}\right) = \log_n\left(n^{-1}\right) = -1
   \]

   Therefore,

   \[
   4 \log_n\left(\frac{1}{n}\right) = 4(-1) = -4
   \]

Now, combining both terms:

\[
\log_{n^2}(n^2) - 4 \log_n\left(\frac{1}{n}\right) = 1 - (-4) = 1 + 4 = 5
\]

Thus, the value of the expression is \( \boxed{5} \).

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. What is the general formula for changing the base of a logarithm?
2. How do logarithmic properties simplify expressions like \(\log_n\left(\frac{1}{n^2}\right)\)?
3. How would the expression change if \( n < 1 \)?
4. What happens if we replace \( n \) with a different base, say \( n^3 \), in the original expression?
5. Can you derive a similar expression using natural logarithms (\(\ln\)) instead?

**Tip:** Always remember that the logarithm of a number with its base is always 1, which can simplify many logarithmic expressions.

Consider the expression \( \log_{n^2}(n^2) - 4 \log_n\left(\frac{1}{n}\right) \) and \( n > 1 \). What is the value of n?

Learn how to simplify the logarithmic expression \( \log_{n^2}(n^2) - 4 \log_n\left(\frac{1}{n}\right) \) with detailed steps. Explore the logarithmic identities and properties used in solving this problem, suitable for students in Grades 10-12.

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