If `log_(2)x xxlog_(2).(x)/(16)+4=0`, then x=

Options

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

Answer

To solve the equation \( \log_{2} x \cdot \frac{\log_{2} x}{16} + 4 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ \frac{(\log_{2} x)^2}{16} + 4 = 0 \]

### Step 2: Isolate the logarithmic term
To isolate the logarithmic term, we can multiply the entire equation by 16 to eliminate the fraction:
\[ (\log_{2} x)^2 + 64 = 0 \]

### Step 3: Rearrange the equation
Now, rearranging the equation gives:
\[ (\log_{2} x)^2 = -64 \]

### Step 4: Analyze the result
Since the square of a real number cannot be negative, this implies that there are no real solutions for \( \log_{2} x \). Therefore, we conclude that there are no real values for \( x \) that satisfy the original equation.

### Conclusion
The equation \( \log_{2} x \cdot \frac{\log_{2} x}{16} + 4 = 0 \) has no real solutions.

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