If `log_(10)x-log_(10)sqrt(x)=2log_(x)10`, then a possible value of x is given by

Options

A. A)10
B. B)100
C. C)1000
D. D)None of these

Answer

To solve the equation \( \log_{10} x - \log_{10} \sqrt{x} = 2 \log_{x} 10 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
We know that \( \sqrt{x} = x^{1/2} \). Therefore, we can rewrite \( \log_{10} \sqrt{x} \) as:
\[ \log_{10} \sqrt{x} = \log_{10} (x^{1/2}) = \frac{1}{2} \log_{10} x \]

### Step 2: Substitute and simplify
Now substitute this back into the original equation:
\[ \log_{10} x - \frac{1}{2} \log_{10} x = 2 \log_{x} 10 \]
This simplifies to:
\[ \frac{1}{2} \log_{10} x = 2 \log_{x} 10 \]

### Step 3: Use the change of base formula
Using the change of base formula, we can express \( \log_{x} 10 \) as:
\[ \log_{x} 10 = \frac{\log_{10} 10}{\log_{10} x} = \frac{1}{\log_{10} x} \]
Substituting this into our equation gives:
\[ \frac{1}{2} \log_{10} x = 2 \cdot \frac{1}{\log_{10} x} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying yields:
\[ \frac{1}{2} (\log_{10} x)^2 = 2 \]

### Step 5: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ (\log_{10} x)^2 = 4 \]

### Step 6: Take the square root
Taking the square root of both sides gives:
\[ \log_{10} x = 2 \quad \text{or} \quad \log_{10} x = -2 \]

### Step 7: Solve for x
1. If \( \log_{10} x = 2 \):
\[ x = 10^2 = 100 \]
2. If \( \log_{10} x = -2 \):
\[ x = 10^{-2} = 0.01 \]

### Conclusion
The possible values of \( x \) are \( 100 \) and \( 0.01 \). Since we are looking for a possible value of \( x \) from the options given (10, 100, 1000, none of these), the answer is:
\[ \boxed{100} \]