If `(1)/("log"_(x)10) = (2)/("log"_(a)10)-2`, then x =

Options

A. `(a)/(2)`
B. `(a)/(100)`
C. `(a^(2))/(10)`
D. `(a^(2))/(100)`

Answer

To solve the equation \( \frac{1}{\log_{x} 10} = \frac{2}{\log_{a} 10} - 2 \), we will follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula.
Using the change of base formula, we can rewrite the logarithms:
\[ \log_{x} 10 = \frac{\log 10}{\log x} \quad \text{and} \quad \log_{a} 10 = \frac{\log 10}{\log a} \]
Substituting these into the equation gives:
\[ \frac{1}{\frac{\log 10}{\log x}} = \frac{2}{\frac{\log 10}{\log a}} - 2 \]

### Step 2: Simplify the left-hand side.
The left-hand side simplifies to:
\[ \frac{\log x}{\log 10} \]
Thus, the equation becomes:
\[ \frac{\log x}{\log 10} = \frac{2 \log a}{\log 10} - 2 \]

### Step 3: Clear the fractions by multiplying through by \(\log 10\).
Multiplying both sides by \(\log 10\) gives:
\[ \log x = 2 \log a - 2 \log 10 \]

### Step 4: Use properties of logarithms to combine terms.
Using the property \( \log a - \log b = \log \left(\frac{a}{b}\right) \), we can rewrite the right-hand side:
\[ \log x = \log a^2 - \log 10^2 \]
This simplifies to:
\[ \log x = \log \left(\frac{a^2}{100}\right) \]

### Step 5: Exponentiate both sides to eliminate the logarithm.
Exponentiating both sides gives:
\[ x = \frac{a^2}{100} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ \boxed{\frac{a^2}{100}} \]