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

Answer

To find the value of \( m \), which is the geometric mean of the three expressions given, we can follow these steps:

### Step 1: Define the expressions
Let:
- \( a = \left(\frac{y}{z}\right)^{\log(yz)} \)
- \( b = \left(\frac{z}{x}\right)^{\log(zx)} \)
- \( c = \left(\frac{x}{y}\right)^{\log(xy)} \)

### Step 2: Write the geometric mean
The geometric mean \( m \) of \( a, b, c \) is given by:
\[ m = (abc)^{1/3} \]

### Step 3: Calculate \( abc \)
Now we will calculate \( abc \):
\[ abc = \left(\frac{y}{z}\right)^{\log(yz)} \cdot \left(\frac{z}{x}\right)^{\log(zx)} \cdot \left(\frac{x}{y}\right)^{\log(xy)} \]

### Step 4: Simplify the expression
We can rewrite \( abc \):
\[ abc = \frac{y^{\log(yz)} \cdot z^{\log(zx)} \cdot x^{\log(xy)}}{z^{\log(yz)} \cdot x^{\log(zx)} \cdot y^{\log(xy)}} \]

### Step 5: Expand the logarithms
Using the property of logarithms \( \log(ab) = \log a + \log b \):
- \( \log(yz) = \log y + \log z \)
- \( \log(zx) = \log z + \log x \)
- \( \log(xy) = \log x + \log y \)

### Step 6: Substitute back into the expression
Substituting back, we get:
\[ abc = \frac{y^{\log y + \log z} \cdot z^{\log z + \log x} \cdot x^{\log x + \log y}}{z^{\log y + \log z} \cdot x^{\log z + \log x} \cdot y^{\log x + \log y}} \]

### Step 7: Cancel out terms
We can see that many terms will cancel out:
- \( y^{\log y} \) cancels with \( y^{\log x + \log y} \)
- \( z^{\log z} \) cancels with \( z^{\log y + \log z} \)
- \( x^{\log x} \) cancels with \( x^{\log z + \log x} \)

After cancellation, we find:
\[ abc = 1 \]

### Step 8: Calculate \( m \)
Now substituting back into the formula for \( m \):
\[ m = (abc)^{1/3} = 1^{1/3} = 1 \]

### Final Answer
Thus, the value of \( m \) is:
\[ \boxed{1} \]