The solution set of the equation `x^(log_x(1-x)^2)=9` is

Options

A. `{-2,4}`
B. {4}
C. `{0,-2,-4}`
D. none of these

Explanation

To solve the equation \( x^{\log_x(1-x)^2} = 9 \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ x^{\log_x(1-x)^2} = 9 \]
Using the property of logarithms, we can rewrite the left side:
\[ \log_x(1-x)^2 = 2 \log_x(1-x) \]
So, we have:
\[ x^{2 \log_x(1-x)} = 9 \]

### Step 2: Apply Logarithm
Taking logarithm on both sides (we can use any base, but base 10 or natural log is common):
\[ \log(x^{2 \log_x(1-x)}) = \log(9) \]
Using the property of logarithms \( \log(a^b) = b \log(a) \):
\[ 2 \log_x(1-x) \cdot \log(x) = \log(9) \]

### Step 3: Simplify Using Change of Base Formula
Using the change of base formula \( \log_x(a) = \frac{\log(a)}{\log(x)} \):
\[ 2 \cdot \frac{\log(1-x)}{\log(x)} \cdot \log(x) = \log(9) \]
This simplifies to:
\[ 2 \log(1-x) = \log(9) \]

### Step 4: Exponentiate to Remove Logarithm
Exponentiating both sides gives:
\[ (1-x)^2 = 9 \]

### Step 5: Solve the Quadratic Equation
Taking the square root of both sides:
\[ 1 - x = 3 \quad \text{or} \quad 1 - x = -3 \]
This leads to two equations:
1. \( 1 - x = 3 \) → \( x = -2 \)
2. \( 1 - x = -3 \) → \( x = 4 \)

### Step 6: Determine Valid Solutions
Since \( x \) must be positive (as it is the base of a logarithm), we discard \( x = -2 \). Thus, the only valid solution is:
\[ x = 4 \]

### Final Answer
The solution set of the equation is:
\[ \{ 4 \} \]

---