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The question asks to express \(\log_n \left(\frac{125n}{16}\right)\) in terms of \(x\) and \(y\), given that:

\[
\log_n 4 = x \quad \text{and} \quad \log_n 5 = y.
\]

Let's work through the expression step by step.

### Step 1: Break down the logarithm of a fraction
We can use the property of logarithms that states:
\[
\log_n \left(\frac{a}{b}\right) = \log_n a - \log_n b.
\]

So,
\[
\log_n \left(\frac{125n}{16}\right) = \log_n (125n) - \log_n 16.
\]

### Step 2: Further simplify each term
We can now simplify each of the logarithmic terms.

- \(\log_n (125n)\): Use the property \(\log_n(ab) = \log_n a + \log_n b\). This gives:
  \[
  \log_n (125n) = \log_n 125 + \log_n n = \log_n 125 + 1.
  \]
  
- \(\log_n 16\): Notice that \(16 = 2^4\), and we can rewrite the logarithm using the properties of exponents and logarithms:
  \[
  \log_n 16 = \log_n (2^4) = 4 \log_n 2.
  \]

### Step 3: Express in terms of \(x\) and \(y\)
Now, we can express everything in terms of the given values of \(x = \log_n 4\) and \(y = \log_n 5\).

- \(\log_n 125 = \log_n (5^3) = 3 \log_n 5 = 3y\).
- \(\log_n 4 = x\), but also \(\log_n 2 = \frac{x}{2}\), since \(4 = 2^2\).

Thus:
\[
\log_n 16 = 4 \log_n 2 = 4 \times \frac{x}{2} = 2x.
\]

### Step 4: Final expression
Now substitute everything back into the equation:
\[
\log_n \left(\frac{125n}{16}\right) = (\log_n 125 + 1) - \log_n 16 = (3y + 1) - 2x.
\]

So the final expression is:
\[
\log_n \left(\frac{125n}{16}\right) = 3y + 1 - 2x.
\]

Would you like any further clarifications or details? Here are some related questions to explore:

1. What is the general rule for the logarithm of a product or a quotient?
2. How can we expand expressions with logarithms involving powers?
3. Why is the change of base formula useful for simplifying logarithms?
4. How can logarithms help in solving exponential equations?
5. Can logarithmic properties be applied to bases other than 10 or \(e\)?

**Tip:** Always look out for opportunities to simplify logarithms by recognizing powers and applying known logarithmic properties.

Given that log_n 4 = x and log_n 5 = y, express log_n (125n/16) in terms of x and y.

Learn how to express log_n (125n/16) in terms of x and y by applying logarithmic properties. This solution involves the quotient, product, and power rules of logarithms, making it suitable for students in Grades 10-12.

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