The value of `x` satisfying the equation `|x-1|^(log_(3)x^(2)-2log_(x)9)=(x-1)^(7)` is

Options

A. `3`
B. `9`
C. `27`
D. `81

Answer

To solve the equation \( |x-1|^{(\log_3 x^2 - 2 \log_x 9)} = (x-1)^7 \), we can follow these steps:

### Step 1: Analyze the Absolute Value
Since we have an absolute value, we need to consider two cases based on the value of \( x \).

**Case 1:** \( x - 1 \geq 0 \) (i.e., \( x \geq 1 \))

In this case, we can drop the absolute value:
\[ (x-1)^{(\log_3 x^2 - 2 \log_x 9)} = (x-1)^7 \]

**Case 2:** \( x - 1 < 0 \) (i.e., \( x < 1 \))

Here, we have:
\[ -(x-1)^{(\log_3 x^2 - 2 \log_x 9)} = (x-1)^7 \]
However, since \( (x-1)^7 \) is negative for \( x < 1 \), this case is not valid.

### Step 2: Simplify the Equation
From Case 1, we can equate the exponents since the bases are the same:
\[ \log_3 x^2 - 2 \log_x 9 = 7 \]

### Step 3: Rewrite the Logarithms
Using the properties of logarithms, we can rewrite \( \log_x 9 \):
\[ \log_x 9 = \frac{\log_3 9}{\log_3 x} = \frac{2}{\log_3 x} \]
Thus, substituting this back into the equation gives us:
\[ \log_3 x^2 - 2 \left(\frac{2}{\log_3 x}\right) = 7 \]

### Step 4: Substitute \( t = \log_3 x \)
Let \( t = \log_3 x \). Then, we have:
\[ 2t - \frac{4}{t} = 7 \]

### Step 5: Clear the Fraction
Multiply through by \( t \) (assuming \( t \neq 0 \)):
\[ 2t^2 - 4 = 7t \]
Rearranging gives:
\[ 2t^2 - 7t - 4 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2} \]
\[ t = \frac{7 \pm \sqrt{49 + 32}}{4} \]
\[ t = \frac{7 \pm \sqrt{81}}{4} \]
\[ t = \frac{7 \pm 9}{4} \]

Calculating the two possible values for \( t \):
1. \( t = \frac{16}{4} = 4 \)
2. \( t = \frac{-2}{4} = -\frac{1}{2} \)

### Step 7: Convert Back to \( x \)
Now, we convert back to \( x \):
1. For \( t = 4 \):
\[ \log_3 x = 4 \implies x = 3^4 = 81 \]
2. For \( t = -\frac{1}{2} \):
\[ \log_3 x = -\frac{1}{2} \implies x = 3^{-\frac{1}{2}} = \frac{1}{\sqrt{3}} \approx 0.577 \]

### Step 8: Determine Valid Solutions
Since we are considering \( x \geq 1 \), we discard \( x = \frac{1}{\sqrt{3}} \).

Thus, the only valid solution is:
\[ \boxed{81} \]