If `xy^(2) = 4 and log_(3) (log_(2) x) + log_(1//3) (log_(1//2) y)=1` , then x equals

Options

A. 4
B. 8
C. 16
D. 64

Explanation

To solve the problem step by step, we will follow the given equations and properties of logarithms.

### Given:
1. \( xy^2 = 4 \)
2. \( \log_3(\log_2 x) + \log_{1/3}(\log_{1/2} y) = 1 \)

### Step 1: Rewrite the logarithmic equation
Using the property of logarithms that states \( \log_{a}(b) = -\log_{1/a}(b) \), we can rewrite the second term:
\[ \log_{1/3}(\log_{1/2} y) = -\log_3(\log_{1/2} y) \]
Thus, we can rewrite the equation as:
\[ \log_3(\log_2 x) - \log_3(\log_{1/2} y) = 1 \]

### Step 2: Combine the logarithms
Using the property \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \):
\[ \log_3\left(\frac{\log_2 x}{\log_{1/2} y}\right) = 1 \]

### Step 3: Exponentiate to eliminate the logarithm
Using the property \( \log_a(b) = c \implies b = a^c \):
\[ \frac{\log_2 x}{\log_{1/2} y} = 3 \]

### Step 4: Rewrite \( \log_{1/2} y \)
Using the property \( \log_{1/2} y = \frac{\log_2 y}{\log_2(1/2)} = -\log_2 y \):
\[ \frac{\log_2 x}{-\log_2 y} = 3 \implies \log_2 x = -3 \log_2 y \]

### Step 5: Express \( x \) in terms of \( y \)
From the equation \( \log_2 x = -3 \log_2 y \):
\[ \log_2 x = \log_2(y^{-3}) \implies x = y^{-3} \]

### Step 6: Substitute \( x \) into the first equation
Substituting \( x = y^{-3} \) into the equation \( xy^2 = 4 \):
\[ (y^{-3})y^2 = 4 \implies y^{-1} = 4 \implies y = \frac{1}{4} \]

### Step 7: Find \( x \)
Now substituting \( y = \frac{1}{4} \) back into \( x = y^{-3} \):
\[ x = \left(\frac{1}{4}\right)^{-3} = 4^3 = 64 \]

### Final Answer:
Thus, the value of \( x \) is \( 64 \).

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