The value of `sqrt((log_(05)^(2)4))` is

Options

A. `-2`
B. `sqrt((-4))`
C. `2`
D. none of these

Answer

To solve the expression \( \sqrt{\log_{0.5}(4)} \), we can follow these steps:

### Step 1: Rewrite the logarithm
We know that \( 0.5 \) can be expressed as \( \frac{1}{2} \). Therefore, we can rewrite the logarithm:
\[ \log_{0.5}(4) = \log_{\frac{1}{2}}(4) \]

### Step 2: Use the change of base formula
Using the change of base formula, we can express the logarithm in terms of base 2:
\[ \log_{\frac{1}{2}}(4) = \frac{\log_{2}(4)}{\log_{2}(\frac{1}{2})} \]

### Step 3: Calculate \( \log_{2}(4) \)
Since \( 4 = 2^2 \), we can find:
\[ \log_{2}(4) = 2 \]

### Step 4: Calculate \( \log_{2}(\frac{1}{2}) \)
We know that \( \frac{1}{2} = 2^{-1} \), so:
\[ \log_{2}(\frac{1}{2}) = -1 \]

### Step 5: Substitute back into the equation
Now we can substitute these values back into our expression:
\[ \log_{\frac{1}{2}}(4) = \frac{2}{-1} = -2 \]

### Step 6: Find the square root
Now we need to find the square root of this value:
\[ \sqrt{\log_{0.5}(4)} = \sqrt{-2} \]

### Step 7: Conclusion
Since the square root of a negative number is not defined in the set of real numbers, we conclude that:
\[ \sqrt{\log_{0.5}(4)} = \sqrt{-2} \text{ (not a real number)} \]