Find the value of `(x^(log y - log z))(y^(log z-log x))(z^(log x - log y)) `

Options

A. xyz
B. `x + y + z`
C. 1
D. 0

Answer

To find the value of the expression \( (x^{\log y - \log z})(y^{\log z - \log x})(z^{\log x - \log y}) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ P = (x^{\log y - \log z})(y^{\log z - \log x})(z^{\log x - \log y}) \]

### Step 2: Apply logarithmic properties
Using the property of logarithms that states \( \log a - \log b = \log \frac{a}{b} \), we can rewrite the exponents:
\[ P = x^{\log \frac{y}{z}} \cdot y^{\log \frac{z}{x}} \cdot z^{\log \frac{x}{y}} \]

### Step 3: Take the logarithm of both sides
Now, we take the logarithm of \( P \):
\[ \log P = \log \left( x^{\log \frac{y}{z}} \cdot y^{\log \frac{z}{x}} \cdot z^{\log \frac{x}{y}} \right) \]
Using the property of logarithms that states \( \log(a \cdot b) = \log a + \log b \):
\[ \log P = \log \left( x^{\log \frac{y}{z}} \right) + \log \left( y^{\log \frac{z}{x}} \right) + \log \left( z^{\log \frac{x}{y}} \right) \]

### Step 4: Simplify using logarithmic identities
Using the property \( \log(a^b) = b \cdot \log a \):
\[ \log P = \left( \log \frac{y}{z} \cdot \log x \right) + \left( \log \frac{z}{x} \cdot \log y \right) + \left( \log \frac{x}{y} \cdot \log z \right) \]

### Step 5: Expand the expression
Now we expand each term:
\[ \log P = \log y \cdot \log x - \log z \cdot \log x + \log z \cdot \log y - \log x \cdot \log y + \log x \cdot \log z - \log y \cdot \log z \]

### Step 6: Combine like terms
Notice that each pair of terms cancels out:
\[ \log P = 0 \]

### Step 7: Conclude the value of \( P \)
Since \( \log P = 0 \), we have:
\[ P = 10^0 = 1 \]

Thus, the value of the expression \( (x^{\log y - \log z})(y^{\log z - \log x})(z^{\log x - \log y}) \) is:
\[ \boxed{1} \]