If `log_(2)x xx log_(2)"" (x)/(16) + 4 = 0`, then x is equal to

Options

A. 4
B. `-4`
C. `1//4`
D. 2

Answer

To solve the equation \( \log_2 x \cdot \log_2 \left( \frac{x}{16} \right) + 4 = 0 \), we can follow these steps:

### Step 1: Rewrite the logarithmic expression
We start by rewriting the second logarithmic term using the property of logarithms:
\[ \log_2 \left( \frac{x}{16} \right) = \log_2 x - \log_2 16 \]
Since \( 16 = 2^4 \), we have:
\[ \log_2 16 = 4 \]
Thus, we can rewrite the equation as:
\[ \log_2 x \cdot \left( \log_2 x - 4 \right) + 4 = 0 \]

### Step 2: Let \( m = \log_2 x \)
Now, let \( m = \log_2 x \). The equation becomes:
\[ m \cdot (m - 4) + 4 = 0 \]
Expanding this gives:
\[ m^2 - 4m + 4 = 0 \]

### Step 3: Factor the quadratic equation
We can factor the quadratic equation:
\[ (m - 2)^2 = 0 \]
This implies:
\[ m - 2 = 0 \quad \Rightarrow \quad m = 2 \]

### Step 4: Solve for \( x \)
Since \( m = \log_2 x \), we have:
\[ \log_2 x = 2 \]
To find \( x \), we rewrite this in exponential form:
\[ x = 2^2 = 4 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{4} \]