`loga^(n)//b^(n)+logb^(n)//c^(n)+llogc^(n)//a^(n)`

Options

A. 1
B. n
C. 0
D. 2

Answer

To solve the expression \( \log \frac{a^n}{b^n} + \log \frac{b^n}{c^n} + \log \frac{c^n}{a^n} \), we can use the properties of logarithms. Here are the steps to find the value:

### Step 1: Apply the Logarithm Property
Using the property of logarithms that states \( \log m + \log n = \log (m \cdot n) \), we can combine the three logarithmic terms:
\[ \log \frac{a^n}{b^n} + \log \frac{b^n}{c^n} + \log \frac{c^n}{a^n} = \log \left( \frac{a^n}{b^n} \cdot \frac{b^n}{c^n} \cdot \frac{c^n}{a^n} \right) \]

### Step 2: Simplify the Expression Inside the Logarithm
Now, we simplify the expression inside the logarithm:
\[ \frac{a^n}{b^n} \cdot \frac{b^n}{c^n} \cdot \frac{c^n}{a^n} = \frac{a^n \cdot b^n \cdot c^n}{b^n \cdot c^n \cdot a^n} \]
Notice that the \( b^n \), \( c^n \), and \( a^n \) in the numerator and denominator will cancel out:
\[ = \frac{a^n \cdot b^n \cdot c^n}{a^n \cdot b^n \cdot c^n} = 1 \]

### Step 3: Apply the Logarithm of 1
Now we have:
\[ \log(1) \]
According to the properties of logarithms, \( \log(1) = 0 \).

### Conclusion
Thus, the value of the original expression is:
\[ \log \frac{a^n}{b^n} + \log \frac{b^n}{c^n} + \log \frac{c^n}{a^n} = 0 \]

### Final Answer
The answer is \( 0 \).

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