`("log"a^3/(bc)+"log"b^3/(ac)+"log"c^3/(ab))` is equal to :

Options

A. 1
B. log abc
C. abc
D. none of these

Answer

To solve the expression \( \log \frac{a^3}{bc} + \log \frac{b^3}{ac} + \log \frac{c^3}{ab} \), we will use the properties of logarithms step by step.

### Step 1: Apply the Logarithm Property
Using the property of logarithms that states \( \log \frac{x}{y} = \log x - \log y \), we can rewrite each term in the expression:

\[ \log \frac{a^3}{bc} = \log a^3 - \log (bc) = 3 \log a - (\log b + \log c) \]

\[ \log \frac{b^3}{ac} = \log b^3 - \log (ac) = 3 \log b - (\log a + \log c) \]

\[ \log \frac{c^3}{ab} = \log c^3 - \log (ab) = 3 \log c - (\log a + \log b) \]

### Step 2: Combine the Terms
Now, we can combine all the terms:

\[ (3 \log a - \log b - \log c) + (3 \log b - \log a - \log c) + (3 \log c - \log a - \log b) \]

### Step 3: Group the Terms
Grouping the logarithmic terms together, we have:

\[ (3 \log a + 3 \log b + 3 \log c) - (\log a + \log b + \log c + \log a + \log b + \log c) \]

This simplifies to:

\[ 3 \log a + 3 \log b + 3 \log c - 2(\log a + \log b + \log c) \]

### Step 4: Simplify Further
Let \( S = \log a + \log b + \log c \). Therefore, we can rewrite the expression as:

\[ 3S - 2S = S \]

### Step 5: Final Expression
Thus, we have:

\[ \log a + \log b + \log c \]

Using the property of logarithms that states \( \log x + \log y + \log z = \log (xyz) \), we can express the final result as:

\[ \log (abc) \]

### Final Answer
The final answer is:

\[ \log (abc) \]

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