`(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 \( \log \left( \frac{a^3}{bc} \right) + \log \left( \frac{b^3}{ac} \right) + \log \left( \frac{c^3}{ab} \right) \), we can follow these steps:

### Step 1: Use the property of logarithms
We know that \( \log(x) + \log(y) = \log(xy) \). Therefore, we can combine the logarithmic expressions:

\[ \log \left( \frac{a^3}{bc} \right) + \log \left( \frac{b^3}{ac} \right) + \log \left( \frac{c^3}{ab} \right) = \log \left( \left( \frac{a^3}{bc} \right) \cdot \left( \frac{b^3}{ac} \right) \cdot \left( \frac{c^3}{ab} \right) \right) \]

### Step 2: Simplify the product inside the logarithm
Now we need to simplify the product:

\[ \frac{a^3}{bc} \cdot \frac{b^3}{ac} \cdot \frac{c^3}{ab} \]

This can be rewritten as:

\[ \frac{a^3 \cdot b^3 \cdot c^3}{(bc)(ac)(ab)} \]

### Step 3: Expand the denominator
The denominator can be expanded:

\[ (bc)(ac)(ab) = b^2c^2a^2 \]

### Step 4: Combine the fractions
Now we can combine the fractions:

\[ \frac{a^3 \cdot b^3 \cdot c^3}{a^2b^2c^2} = \frac{a^{3-2} \cdot b^{3-2} \cdot c^{3-2}}{1} = abc \]

### Step 5: Write the final logarithmic expression
Now we can write:

\[ \log \left( abc \right) \]

### Final Answer
Thus, the final result is:

\[ \log(abc) \]