If `x^((3)/(2)("log"_(2) x-3)) = (1)/(8)`, then x equals to

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the equation \( x^{\frac{3}{2}(\log_2 x - 3)} = \frac{1}{8} \), we will follow these steps:

### Step 1: Rewrite the right-hand side
We know that \( \frac{1}{8} \) can be expressed as \( 2^{-3} \):
\[ x^{\frac{3}{2}(\log_2 x - 3)} = 2^{-3} \]

### Step 2: Take logarithm on both sides
Taking logarithm base 2 on both sides:
\[ \log_2\left(x^{\frac{3}{2}(\log_2 x - 3)}\right) = \log_2(2^{-3}) \]

### Step 3: Apply the logarithm power rule
Using the power rule of logarithms, we can simplify the left-hand side:
\[ \frac{3}{2}(\log_2 x - 3) \cdot \log_2 x = -3 \]

### Step 4: Distribute and simplify
Expanding the left-hand side:
\[ \frac{3}{2}(\log_2^2 x - 3\log_2 x) = -3 \]
Multiplying through by \( \frac{2}{3} \) to eliminate the fraction:
\[ \log_2^2 x - 3\log_2 x = -2 \]

### Step 5: Rearrange into standard quadratic form
Rearranging gives us:
\[ \log_2^2 x - 3\log_2 x + 2 = 0 \]

### Step 6: Factor the quadratic equation
Factoring the quadratic:
\[ (\log_2 x - 1)(\log_2 x - 2) = 0 \]

### Step 7: Solve for \( \log_2 x \)
Setting each factor to zero gives us two possible solutions:
1. \( \log_2 x - 1 = 0 \) → \( \log_2 x = 1 \) → \( x = 2^1 = 2 \)
2. \( \log_2 x - 2 = 0 \) → \( \log_2 x = 2 \) → \( x = 2^2 = 4 \)

### Step 8: Conclusion
The possible values for \( x \) are \( 2 \) and \( 4 \).