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 equation
We start with the given equation:
\[ \frac{1}{\log_{x}10} = \frac{2}{\log_{a}10} - 2 \]

### Step 2: Simplify the right-hand side
To simplify the right-hand side, we can express \( 2 \) in terms of logarithms:
\[ 2 = \frac{2 \log_{a}10}{\log_{a}10} \]
Thus, we rewrite the equation as:
\[ \frac{1}{\log_{x}10} = \frac{2 - 2 \log_{a}10}{\log_{a}10} \]
This simplifies to:
\[ \frac{1}{\log_{x}10} = \frac{2 - 2 \log_{a}10}{\log_{a}10} \]

### Step 3: Combine the fractions on the right-hand side
Now we can combine the fractions on the right:
\[ \frac{1}{\log_{x}10} = \frac{2 - 2 \log_{a}10}{\log_{a}10} = \frac{2(1 - \log_{a}10)}{\log_{a}10} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ \log_{a}10 = 2(1 - \log_{a}10) \log_{x}10 \]

### Step 5: Distribute and rearrange
Distributing on the right-hand side:
\[ \log_{a}10 = 2 \log_{x}10 - 2 \log_{x}10 \log_{a}10 \]
Rearranging the equation:
\[ \log_{a}10 + 2 \log_{x}10 \log_{a}10 = 2 \log_{x}10 \]

### Step 6: Factor out common terms
Factoring out \( \log_{a}10 \):
\[ \log_{a}10 (1 + 2 \log_{x}10) = 2 \log_{x}10 \]

### Step 7: Isolate \( \log_{x}10 \)
Now we can isolate \( \log_{x}10 \):
\[ \log_{x}10 = \frac{\log_{a}10}{2 \log_{a}10 - 1} \]

### Step 8: Convert back to exponential form
Using the property of logarithms, we can express \( x \) in terms of \( a \):
\[ x = 10^{\frac{\log_{a}10}{2 \log_{a}10 - 1}} \]

### Step 9: Simplify the expression
Using the change of base formula, we can express this as:
\[ x = \frac{a^2}{100} \]

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

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