The solution set of the equation
`"log"_(x)2 xx "log"_(2x)2 = "log"_(4x) 2,` is

Options

A. `{2^(-sqrt(2)), 2^(sqrt(2))}`
B. `{(1)/(2), 2}`
C. `{(1)/(4), 4}`
D. none of these

Explanation

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

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express the logarithms as:
\[ \log_{x} 2 = \frac{\log 2}{\log x}, \quad \log_{2x} 2 = \frac{\log 2}{\log(2x)} = \frac{\log 2}{\log 2 + \log x}, \quad \log_{4x} 2 = \frac{\log 2}{\log(4x)} = \frac{\log 2}{\log 4 + \log x} = \frac{\log 2}{2\log 2 + \log x} \]

### Step 2: Substitute these values into the equation
Substituting these values into the original equation gives:
\[ \frac{\log 2}{\log x} \cdot \frac{\log 2}{\log 2 + \log x} = \frac{\log 2}{2\log 2 + \log x} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying yields:
\[ \log 2 \cdot \log 2 = \log x \cdot \frac{\log 2}{2\log 2 + \log x} \cdot (\log 2 + \log x) \]
This simplifies to:
\[ (\log 2)^2 = \log x \cdot \log 2 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ (\log 2)^2 = \log 2 \cdot \log x \]
Assuming \( \log 2 \neq 0 \) (which is true), we can divide both sides by \( \log 2 \):
\[ \log 2 = \log x \]

### Step 5: Solve for \( x \)
From \( \log 2 = \log x \), we have:
\[ x = 2 \]

### Step 6: Check for other solutions
Now, we need to check if there are any other solutions. We also consider the case where:
\[ \log x = 2 \cdot \log 2 \quad \Rightarrow \quad x = 2^2 = 4 \]

### Step 7: Final solution set
Thus, the solution set for the equation is:
\[ \{2, 4\} \]