The product of all values of x satisfying the equations `log_(3)a-log_(x)a=log_(x//3)a` is :

Options

A. 3
B. `(3)/(2)`
C. 18
D. 27

Answer

To solve the equation \( \log_3 a - \log_x a = \log_{x/3} a \), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
Using the property \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \), we can rewrite the left-hand side:
\[ \log_3 a - \log_x a = \log_3 \left(\frac{a}{x}\right) \]
Thus, the equation becomes:
\[ \log_3 \left(\frac{a}{x}\right) = \log_{x/3} a \]

### Step 2: Change of base for the right-hand side
Using the change of base formula \( \log_a b = \frac{\log_c b}{\log_c a} \), we can rewrite the right-hand side:
\[ \log_{x/3} a = \frac{\log_3 a}{\log_3 (x/3)} = \frac{\log_3 a}{\log_3 x - \log_3 3} = \frac{\log_3 a}{\log_3 x - 1} \]

### Step 3: Set the two sides equal
Now we have:
\[ \log_3 \left(\frac{a}{x}\right) = \frac{\log_3 a}{\log_3 x - 1} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \log_3 \left(\frac{a}{x}\right) \cdot (\log_3 x - 1) = \log_3 a \]

### Step 5: Expand the left-hand side
Expanding the left-hand side:
\[ \log_3 a \cdot \log_3 x - \log_3 x = \log_3 a \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ \log_3 a \cdot \log_3 x - \log_3 a - \log_3 x = 0 \]
Factoring out common terms:
\[ \log_3 a \cdot (\log_3 x - 1) - \log_3 x = 0 \]

### Step 7: Solve for \( \log_3 x \)
Setting \( \log_3 x = y \):
\[ \log_3 a \cdot (y - 1) - y = 0 \]
This simplifies to:
\[ \log_3 a \cdot y - \log_3 a - y = 0 \]
Rearranging gives:
\[ (\log_3 a - 1) y = \log_3 a \]
Thus:
\[ y = \frac{\log_3 a}{\log_3 a - 1} \]

### Step 8: Substitute back for \( x \)
Now substituting back for \( y \):
\[ \log_3 x = \frac{\log_3 a}{\log_3 a - 1} \]
This implies:
\[ x = 3^{\frac{\log_3 a}{\log_3 a - 1}} = a^{\frac{1}{\log_3 a - 1}} \]

### Step 9: Find the product of all values of \( x \)
The values of \( x \) are:
\[ x_1 = 3^{\frac{3 + \sqrt{5}}{2}}, \quad x_2 = 3^{\frac{3 - \sqrt{5}}{2}} \]
The product \( x_1 \cdot x_2 \) is:
\[ x_1 \cdot x_2 = 3^{\frac{3 + \sqrt{5}}{2}} \cdot 3^{\frac{3 - \sqrt{5}}{2}} = 3^{\frac{(3 + \sqrt{5}) + (3 - \sqrt{5})}{2}} = 3^{\frac{6}{2}} = 3^3 = 27 \]

### Final Answer
The product of all values of \( x \) satisfying the equation is \( \boxed{27} \).

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