The value of `x^((logy-logz))xx y^((logz - logx)) xx z^((log x- logy))` is equal to

Options

A. 1
B. 3
C. 0
D. 5

Answer

To solve the expression \( x^{(\log y - \log z)} \cdot y^{(\log z - \log x)} \cdot z^{(\log x - \log y)} \), we will follow these steps:

### Step 1: Rewrite the expression using properties of logarithms
We can rewrite the logarithmic differences using the property that \( \log a - \log b = \log \left( \frac{a}{b} \right) \).

So, we have:
\[ x^{(\log y - \log z)} = x^{\log \left( \frac{y}{z} \right)} \]
\[ y^{(\log z - \log x)} = y^{\log \left( \frac{z}{x} \right)} \]
\[ z^{(\log x - \log y)} = z^{\log \left( \frac{x}{y} \right)} \]

### Step 2: Combine the expression
Now, substituting these back into the original expression gives us:
\[ x^{\log \left( \frac{y}{z} \right)} \cdot y^{\log \left( \frac{z}{x} \right)} \cdot z^{\log \left( \frac{x}{y} \right)} \]

### Step 3: Use the property of exponents
Using the property \( a^{\log_b c} = c^{\log_b a} \), we can rewrite each term:
\[ x^{\log \left( \frac{y}{z} \right)} = \left( \frac{y}{z} \right)^{\log x} \]
\[ y^{\log \left( \frac{z}{x} \right)} = \left( \frac{z}{x} \right)^{\log y} \]
\[ z^{\log \left( \frac{x}{y} \right)} = \left( \frac{x}{y} \right)^{\log z} \]

### Step 4: Combine all the rewritten terms
Now, substituting these back, we have:
\[ \left( \frac{y}{z} \right)^{\log x} \cdot \left( \frac{z}{x} \right)^{\log y} \cdot \left( \frac{x}{y} \right)^{\log z} \]

### Step 5: Simplify the expression
This can be simplified as follows:
\[ = \frac{y^{\log x}}{z^{\log x}} \cdot \frac{z^{\log y}}{x^{\log y}} \cdot \frac{x^{\log z}}{y^{\log z}} \]

### Step 6: Combine the fractions
Combining the fractions gives:
\[ = \frac{y^{\log x} \cdot z^{\log y} \cdot x^{\log z}}{z^{\log x} \cdot x^{\log y} \cdot y^{\log z}} \]

### Step 7: Notice the cancellation
Now, we can see that each base appears in both the numerator and the denominator:
- \( y^{\log x} \) cancels with \( y^{\log z} \)
- \( z^{\log y} \) cancels with \( z^{\log x} \)
- \( x^{\log z} \) cancels with \( x^{\log y} \)

Thus, we are left with:
\[ = 1 \]

### Conclusion
The value of the expression \( x^{(\log y - \log z)} \cdot y^{(\log z - \log x)} \cdot z^{(\log x - \log y)} \) is equal to \( 1 \).