If `x_(1)` and `x_(2)` are solution of the equation `log_(5)(log_(64)|x|+(25)^(x)-(1)/(2))=2x`, then

Options

A. `x_(1)=2x_(2)`
B. `x_(1)+x_(2)=0`
C. `x_(1)=3x_(2)`
D. `x_(1)x_(2)=64`

Explanation

To solve the equation \( \log_{5} \left( \log_{64} |x| + 25^{x} - \frac{1}{2} \right) = 2x \), we will follow these steps:

### Step 1: Rewrite the Logarithmic Equation
Using the property of logarithms, if \( \log_{a}(y) = x \), then \( y = a^{x} \). Here, we can rewrite the equation as:
\[ \log_{64} |x| + 25^{x} - \frac{1}{2} = 5^{2x} \]

### Step 2: Isolate the Logarithmic Term
Rearranging the equation gives:
\[ \log_{64} |x| = 5^{2x} - 25^{x} + \frac{1}{2} \]

### Step 3: Express \( 25^{x} \) in Terms of \( 5^{x} \)
Since \( 25 = 5^{2} \), we can express \( 25^{x} \) as \( (5^{2})^{x} = 5^{2x} \). Thus, we can rewrite the equation as:
\[ \log_{64} |x| = 5^{2x} - 5^{2x} + \frac{1}{2} \]
This simplifies to:
\[ \log_{64} |x| = \frac{1}{2} \]

### Step 4: Solve for \( |x| \)
Using the property of logarithms again, we can rewrite this as:
\[ |x| = 64^{\frac{1}{2}} = \sqrt{64} = 8 \]

### Step 5: Find the Values of \( x \)
Since we have \( |x| = 8 \), this gives us two possible solutions:
\[ x = 8 \quad \text{or} \quad x = -8 \]

### Final Solutions
Thus, the solutions to the equation are:
\[ x_1 = 8 \quad \text{and} \quad x_2 = -8 \]