If `log_(x)log_(18) (sqrt(2)+sqrt(8)) = (-1)/(2)`. Then the value of 'x'.

Options

A. is a perfect square
B. is a composite number
C. relatively prime with 25
D. is a prime number

Answer

To solve the equation \( \log_{x} \left( \log_{18} \left( \sqrt{2} + \sqrt{8} \right) \right) = -\frac{1}{2} \), we will follow these steps:

### Step 1: Simplify the expression inside the logarithm
First, we simplify \( \sqrt{2} + \sqrt{8} \):
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
Thus,
\[ \sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2} \]

### Step 2: Rewrite the equation
Now we can rewrite the original equation as:
\[ \log_{x} \left( \log_{18} (3\sqrt{2}) \right) = -\frac{1}{2} \]

### Step 3: Use properties of logarithms
Using the property of logarithms, we can express the equation as:
\[ \log_{x} \left( \log_{18} (3\sqrt{2}) \right) = \log_{x} \left( \frac{1}{\sqrt{2}} \right) \]

### Step 4: Set the arguments equal
Since the bases are the same, we can set the arguments equal:
\[ \log_{18} (3\sqrt{2}) = \frac{1}{\sqrt{x}} \]

### Step 5: Solve for \( x \)
Now we need to express \( \log_{18} (3\sqrt{2}) \):
\[ 3\sqrt{2} = 3 \cdot 2^{1/2} = 3^{1} \cdot 2^{1/2} \]
Using the change of base formula:
\[ \log_{18} (3\sqrt{2}) = \log_{18} (3) + \log_{18} (2^{1/2}) = \log_{18} (3) + \frac{1}{2} \log_{18} (2) \]

### Step 6: Substitute back into the equation
Substituting this back, we have:
\[ \log_{18} (3) + \frac{1}{2} \log_{18} (2) = \frac{1}{\sqrt{x}} \]

### Step 7: Isolate \( x \)
To isolate \( x \), we take the reciprocal:
\[ \sqrt{x} = \frac{1}{\log_{18} (3) + \frac{1}{2} \log_{18} (2)} \]
Squaring both sides gives:
\[ x = \left( \frac{1}{\log_{18} (3) + \frac{1}{2} \log_{18} (2)} \right)^{2} \]

### Step 8: Calculate the value of \( x \)
To find the numerical value, we can compute \( \log_{18} (3) \) and \( \log_{18} (2) \) using the change of base formula:
\[ \log_{18} (3) = \frac{\log_{10} (3)}{\log_{10} (18)}, \quad \log_{18} (2) = \frac{\log_{10} (2)}{\log_{10} (18)} \]
After calculating these values, we can substitute them back to find \( x \).

### Final Answer
After performing the calculations, we find that:
\[ x = 4 \]