the number of roots of the equation `log_(3sqrtx) x + log_(3x) (sqrtx) =0` is

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To solve the equation \( \log_{3\sqrt{x}} x + \log_{3x} (\sqrt{x}) = 0 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
Using the change of base formula, we can rewrite the logarithmic expressions:
\[ \log_{3\sqrt{x}} x = \frac{\log x}{\log (3\sqrt{x})} = \frac{\log x}{\log 3 + \frac{1}{2} \log x} \]
\[ \log_{3x} (\sqrt{x}) = \frac{\log (\sqrt{x})}{\log (3x)} = \frac{\frac{1}{2} \log x}{\log 3 + \log x} \]

### Step 2: Substitute back into the equation
Now substituting these back into the equation, we have:
\[ \frac{\log x}{\log 3 + \frac{1}{2} \log x} + \frac{\frac{1}{2} \log x}{\log 3 + \log x} = 0 \]

### Step 3: Combine the fractions
To combine the fractions, we need a common denominator:
\[ \frac{\log x (\log 3 + \log x) + \frac{1}{2} \log x (\log 3 + \frac{1}{2} \log x)}{(\log 3 + \frac{1}{2} \log x)(\log 3 + \log x)} = 0 \]
The numerator must equal zero:
\[ \log x (\log 3 + \log x) + \frac{1}{2} \log x (\log 3 + \frac{1}{2} \log x) = 0 \]

### Step 4: Factor out \(\log x\)
Factoring out \(\log x\) gives:
\[ \log x \left( \log 3 + \log x + \frac{1}{2} (\log 3 + \frac{1}{2} \log x) \right) = 0 \]
This implies:
1. \( \log x = 0 \) which gives \( x = 1 \)
2. The other factor must also equal zero.

### Step 5: Solve the remaining equation
Now we simplify the remaining equation:
\[ \log 3 + \log x + \frac{1}{2} \log 3 + \frac{1}{4} \log x = 0 \]
Combining like terms:
\[ \left(1 + \frac{1}{4}\right) \log x + \left(1 + \frac{1}{2}\right) \log 3 = 0 \]
This simplifies to:
\[ \frac{5}{4} \log x + \frac{3}{2} \log 3 = 0 \]
Rearranging gives:
\[ \frac{5}{4} \log x = -\frac{3}{2} \log 3 \]
Thus:
\[ \log x = -\frac{3}{2} \cdot \frac{4}{5} \log 3 = -\frac{6}{5} \log 3 \]
Exponentiating both sides:
\[ x = 3^{-\frac{6}{5}} = \frac{1}{3^{\frac{6}{5}}} \]

### Step 6: Conclusion
We have found two solutions:
1. \( x = 1 \)
2. \( x = 3^{-\frac{6}{5}} \)

Both solutions are positive. Therefore, the total number of roots of the equation is **2**.