Find x, if `log x^(3) - log 3x =2 log 2 + log 3`,

Options

A. 3
B. 6
C. 9
D. none of these

Answer

To solve the equation \( \log x^3 - \log 3x = 2 \log 2 + \log 3 \), we will follow a series of steps using logarithmic properties.

### Step-by-Step Solution:

1. **Apply Logarithmic Properties**:
We start with the left-hand side of the equation:
\[ \log x^3 - \log 3x \]
Using the property \( \log a - \log b = \log \frac{a}{b} \), we can rewrite this as:
\[ \log \frac{x^3}{3x} \]

2. **Simplify the Expression**:
Now, simplify \( \frac{x^3}{3x} \):
\[ \frac{x^3}{3x} = \frac{x^2}{3} \]
So, we have:
\[ \log \frac{x^2}{3} \]

3. **Rewrite the Right-Hand Side**:
Now, let's simplify the right-hand side:
\[ 2 \log 2 + \log 3 \]
Using the property \( a \log b = \log b^a \), we can rewrite \( 2 \log 2 \) as \( \log 2^2 = \log 4 \). Therefore, the right-hand side becomes:
\[ \log 4 + \log 3 \]
Using the property \( \log a + \log b = \log (a \cdot b) \), we can combine these:
\[ \log (4 \cdot 3) = \log 12 \]

4. **Set the Two Sides Equal**:
Now we have:
\[ \log \frac{x^2}{3} = \log 12 \]

5. **Eliminate the Logarithm**:
Since the logarithm is equal on both sides, we can equate the arguments:
\[ \frac{x^2}{3} = 12 \]

6. **Solve for \( x^2 \)**:
Multiply both sides by 3 to isolate \( x^2 \):
\[ x^2 = 12 \cdot 3 = 36 \]

7. **Take the Square Root**:
Taking the square root of both sides gives:
\[ x = \pm 6 \]

8. **Consider the Domain**:
Since we are dealing with logarithms, \( x \) must be greater than 0. Thus, we discard the negative solution:
\[ x = 6 \]

### Final Answer:
The value of \( x \) is \( 6 \).