If m is the geometric mean of
`((y)/(z))^(log(yz)),((z)/(x))^(log(zx))"and"((x)/(y))^(log(xy))`
then what is the value of m?

Options

A. 1
B. 3
C. 6
D. 9

Explanation

To find the geometric mean \( m \) of the three given terms:

1. **Identify the terms**:
- Let \( A = \left( \frac{y}{z} \right)^{\log(yz)} \)
- Let \( B = \left( \frac{z}{x} \right)^{\log(zx)} \)
- Let \( C = \left( \frac{x}{y} \right)^{\log(xy)} \)

2. **Calculate the product \( A \cdot B \cdot C \)**:
\[ A \cdot B \cdot C = \left( \frac{y}{z} \right)^{\log(yz)} \cdot \left( \frac{z}{x} \right)^{\log(zx)} \cdot \left( \frac{x}{y} \right)^{\log(xy)} \]

3. **Simplify each term**:
- Rewrite \( A \):
\[ A = \left( \frac{y}{z} \right)^{\log(y) + \log(z)} = \frac{y^{\log(y) + \log(z)}}{z^{\log(y) + \log(z)}} \]
- Rewrite \( B \):
\[ B = \left( \frac{z}{x} \right)^{\log(z) + \log(x)} = \frac{z^{\log(z) + \log(x)}}{x^{\log(z) + \log(x)}} \]
- Rewrite \( C \):
\[ C = \left( \frac{x}{y} \right)^{\log(x) + \log(y)} = \frac{x^{\log(x) + \log(y)}}{y^{\log(x) + \log(y)}} \]

4. **Combine the expressions**:
\[ A \cdot B \cdot C = \frac{y^{\log(y) + \log(z)}}{z^{\log(y) + \log(z)}} \cdot \frac{z^{\log(z) + \log(x)}}{x^{\log(z) + \log(x)}} \cdot \frac{x^{\log(x) + \log(y)}}{y^{\log(x) + \log(y)}} \]

5. **Combine all the numerators and denominators**:
\[ = \frac{y^{\log(y) + \log(z)} \cdot z^{\log(z) + \log(x)} \cdot x^{\log(x) + \log(y)}}{z^{\log(y) + \log(z)} \cdot x^{\log(z) + \log(x)} \cdot y^{\log(x) + \log(y)}} \]

6. **Notice the cancellation**:
- The terms will cancel out, leading to:
\[ = 1 \]

7. **Find the geometric mean**:
\[ m = \sqrt[3]{A \cdot B \cdot C} = \sqrt[3]{1} = 1 \]

Thus, the value of \( m \) is \( 1 \).