`(1)/(log_(xy)xyz)+(1)/(log_(yz)xyz)+(1)/(log_(zx)xyz)` =

Options

A. `0`
B. `1`
C. `2`
D. `log_(x)xyz`

Answer

To solve the expression

\[ \frac{1}{\log_{xy} xyz} + \frac{1}{\log_{yz} xyz} + \frac{1}{\log_{zx} xyz} \]

we can use the change of base formula for logarithms, which states that

\[ \log_{a} b = \frac{\log b}{\log a}. \]

### Step 1: Apply the Change of Base Formula

Using the change of base formula, we can rewrite each term in the expression:

\[ \frac{1}{\log_{xy} xyz} = \frac{\log xy}{\log xyz}, \quad \frac{1}{\log_{yz} xyz} = \frac{\log yz}{\log xyz}, \quad \frac{1}{\log_{zx} xyz} = \frac{\log zx}{\log xyz}. \]

Thus, we can rewrite the entire expression as:

\[ \frac{\log xy}{\log xyz} + \frac{\log yz}{\log xyz} + \frac{\log zx}{\log xyz}. \]

### Step 2: Combine the Terms

Now, we can combine these fractions since they all have the same denominator:

\[ \frac{\log xy + \log yz + \log zx}{\log xyz}. \]

### Step 3: Use the Logarithm Property

Using the property of logarithms that states \(\log a + \log b = \log(ab)\), we can combine the numerator:

\[ \log xy + \log yz + \log zx = \log(xy \cdot yz \cdot zx). \]

### Step 4: Simplify the Product

Now, let's simplify the product \(xy \cdot yz \cdot zx\):

\[ xy \cdot yz \cdot zx = x^2y^2z^2. \]

Thus, we have:

\[ \log(xy \cdot yz \cdot zx) = \log(x^2y^2z^2) = 2\log(xyz). \]

### Step 5: Substitute Back into the Expression

Now substituting back into our expression gives:

\[ \frac{2 \log(xyz)}{\log xyz}. \]

### Step 6: Simplify the Expression

Since \(\frac{\log(xyz)}{\log(xyz)} = 1\), we have:

\[ 2 \cdot 1 = 2. \]

### Final Answer

Thus, the value of the original expression is

\[ \boxed{2}. \]