If `x=-2`, then the value of `"log"_(4)((x^(2))/(4)) -2 "log"_(4)(4x^(4)),` is

Options

A. 2
B. `-4`
C. `-6`
D. 0

Answer

To solve the expression \( \log_{4}\left(\frac{x^{2}}{4}\right) - 2 \log_{4}(4x^{4}) \) when \( x = -2 \), we can follow these steps:

### Step 1: Substitute \( x = -2 \) into the expression
We start by substituting \( x \) with \(-2\):
\[ \log_{4}\left(\frac{(-2)^{2}}{4}\right) - 2 \log_{4}(4(-2)^{4}) \]

### Step 2: Simplify the first logarithm
Calculate \((-2)^{2}\):
\[ (-2)^{2} = 4 \]
Now substitute this back into the logarithm:
\[ \log_{4}\left(\frac{4}{4}\right) = \log_{4}(1) \]

### Step 3: Evaluate \( \log_{4}(1) \)
The logarithm of 1 in any base is 0:
\[ \log_{4}(1) = 0 \]

### Step 4: Simplify the second logarithm
Now we simplify the second part \( 2 \log_{4}(4(-2)^{4}) \):
First, calculate \((-2)^{4}\):
\[ (-2)^{4} = 16 \]
Now substitute this back:
\[ 2 \log_{4}(4 \cdot 16) = 2 \log_{4}(64) \]

### Step 5: Evaluate \( \log_{4}(64) \)
Recognize that \( 64 = 4^{3} \):
\[ \log_{4}(64) = \log_{4}(4^{3}) = 3 \]

### Step 6: Substitute back into the expression
Now substitute this value back into the expression:
\[ 0 - 2 \cdot 3 = 0 - 6 = -6 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{-6} \]

---