The value of `(bc)^log(b/c)*(ca)^log(c/a)*(ab)^log(a/b)` is

Options

A. 0
B. `-1`
C. 1
D. none of these

Explanation

To solve the expression \((bc)^{\log(b/c)} \cdot (ca)^{\log(c/a)} \cdot (ab)^{\log(a/b)}\), we will follow these steps:

### Step 1: Rewrite the expression
Let \( t = (bc)^{\log(b/c)} \cdot (ca)^{\log(c/a)} \cdot (ab)^{\log(a/b)} \).

### Step 2: Take the logarithm of both sides
Taking the logarithm of \( t \):
\[ \log t = \log \left( (bc)^{\log(b/c)} \cdot (ca)^{\log(c/a)} \cdot (ab)^{\log(a/b)} \right) \]

### Step 3: Apply the logarithm properties
Using the property \(\log(xy) = \log x + \log y\):
\[ \log t = \log(bc)^{\log(b/c)} + \log(ca)^{\log(c/a)} + \log(ab)^{\log(a/b)} \]

### Step 4: Simplify using the power rule
Using the property \(\log(x^n) = n \log x\):
\[ \log t = \log(b/c) \cdot \log(bc) + \log(c/a) \cdot \log(ca) + \log(a/b) \cdot \log(ab) \]

### Step 5: Expand each term
Now we expand each logarithm:
- \(\log(bc) = \log b + \log c\)
- \(\log(ca) = \log c + \log a\)
- \(\log(ab) = \log a + \log b\)

Substituting these into the equation:
\[ \log t = \log(b/c)(\log b + \log c) + \log(c/a)(\log c + \log a) + \log(a/b)(\log a + \log b) \]

### Step 6: Expand the products
Now we expand each product:
\[ \log t = \left( \log b - \log c \right)(\log b + \log c) + \left( \log c - \log a \right)(\log c + \log a) + \left( \log a - \log b \right)(\log a + \log b) \]

### Step 7: Collect like terms
After expanding, we will collect like terms:
1. The terms involving \(\log^2 b\), \(\log^2 c\), and \(\log^2 a\).
2. The cross terms involving \(\log a \log b\), \(\log b \log c\), and \(\log c \log a\).

### Step 8: Simplify the expression
After simplification, we will find that all terms cancel out, leading to:
\[ \log t = 0 \]

### Step 9: Solve for \( t \)
Using the property of logarithms, if \(\log t = 0\), then:
\[ t = 10^0 = 1 \]

### Final Answer
Thus, the value of the expression \((bc)^{\log(b/c)} \cdot (ca)^{\log(c/a)} \cdot (ab)^{\log(a/b)}\) is:
\[ \boxed{1} \]