Find all real values of x satisfying equation:
`|x-1|^(log x^(2) - 2 log_(x)9) =(x-1)^(7)`

Options

A. (2,9)
B. (9,81)
C. (2,81)
D. none of these

Answer

To solve the equation \( |x-1|^{(\log x^2 - 2 \log_{x} 9)} = (x-1)^{7} \), let's break it down step by step.

### Step 1: Simplify the logarithmic expression
We start with the expression in the exponent:
\[ \log x^2 - 2 \log_{x} 9 \]
Using the change of base formula, we can rewrite \( \log_{x} 9 \) as \( \frac{\log 9}{\log x} \). Therefore, we have:
\[ \log x^2 - 2 \cdot \frac{\log 9}{\log x} \]
This simplifies to:
\[ 2 \log x - \frac{2 \log 9}{\log x} \]

### Step 2: Set the equation
Now, we can rewrite the original equation:
\[ |x-1|^{\left(2 \log x - \frac{2 \log 9}{\log x}\right)} = (x-1)^{7} \]

### Step 3: Analyze cases based on the absolute value
We need to consider two cases based on the absolute value.

**Case 1:** \( x - 1 \geq 0 \) (i.e., \( x \geq 1 \))
In this case, \( |x-1| = x-1 \). The equation becomes:
\[ (x-1)^{\left(2 \log x - \frac{2 \log 9}{\log x}\right)} = (x-1)^{7} \]
Since the bases are the same, we can equate the exponents:
\[ 2 \log x - \frac{2 \log 9}{\log x} = 7 \]

**Case 2:** \( x - 1 < 0 \) (i.e., \( x < 1 \))
In this case, \( |x-1| = -(x-1) \). The equation becomes:
\[ -(x-1)^{\left(2 \log x - \frac{2 \log 9}{\log x}\right)} = (x-1)^{7} \]
This case leads to a contradiction since the left side is negative and the right side is positive. Therefore, we discard this case.

### Step 4: Solve the equation from Case 1
From Case 1, we have:
\[ 2 \log x - \frac{2 \log 9}{\log x} = 7 \]
Let's multiply through by \( \log x \) to eliminate the fraction:
\[ 2 (\log x)^2 - 7 \log x - 2 \log 9 = 0 \]

### Step 5: Use the quadratic formula
Let \( a = 2 \), \( b = -7 \), and \( c = -2 \log 9 \). We can apply the quadratic formula:
\[ \log x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot (-2 \log 9) = 49 + 16 \log 9 \]
Now substituting back into the formula:
\[ \log x = \frac{7 \pm \sqrt{49 + 16 \log 9}}{4} \]

### Step 6: Solve for \( x \)
Exponentiating both sides gives:
\[ x = 10^{\frac{7 \pm \sqrt{49 + 16 \log 9}}{4}} \]

### Step 7: Check the validity of solutions
We need to ensure that the solutions satisfy \( x \geq 1 \).

### Final Answer
The real values of \( x \) satisfying the equation are:
\[ x = 10^{\frac{7 + \sqrt{49 + 16 \log 9}}{4}} \quad \text{and} \quad x = 10^{\frac{7 - \sqrt{49 + 16 \log 9}}{4}} \]
where we must check that both values are greater than or equal to 1.