The value of `[1/(log_(a//b)x) + 1/(log_(b//c)x) + 1/(log_(c//a) x)]` is:

Options

A. 0
B. 1
C. abc
D. `x^(3)`

Answer

To solve the expression \(\frac{1}{\log_{a/b} x} + \frac{1}{\log_{b/c} x} + \frac{1}{\log_{c/a} x}\), we can use properties of logarithms to simplify it step by step.

### Step 1: Apply the Change of Base Formula
Using the property of logarithms that states \(\log_{a/b} x = \frac{1}{\log_{x} (a/b)}\), we can rewrite each term in the expression:

\[ \frac{1}{\log_{a/b} x} = \log_{x} (a/b) \]
\[ \frac{1}{\log_{b/c} x} = \log_{x} (b/c) \]
\[ \frac{1}{\log_{c/a} x} = \log_{x} (c/a) \]

### Step 2: Combine the Logarithms
Now we can combine these logarithmic terms:

\[ \log_{x} (a/b) + \log_{x} (b/c) + \log_{x} (c/a) \]

Using the property of logarithms that states \(\log_{x} m + \log_{x} n = \log_{x} (m \cdot n)\), we can combine the logs:

\[ \log_{x} \left( \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} \right) \]

### Step 3: Simplify the Argument
Now, simplify the argument of the logarithm:

\[ \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} = \frac{a \cdot b \cdot c}{b \cdot c \cdot a} = 1 \]

### Step 4: Evaluate the Logarithm
Now we have:

\[ \log_{x} (1) \]

Using the property of logarithms that states \(\log_{b} (1) = 0\) for any base \(b\), we find:

\[ \log_{x} (1) = 0 \]

### Final Answer
Thus, the value of the original expression is:

\[ \boxed{0} \]