Find the values of x satisfying the equation `|x-2|^(log_(3)x^(4)-3log_(x)9)(x-2)^(10)=1`.

Answer

To solve the equation \( |x-2|^{(\log_{3}(x^{4}) - 3\log_{x}(9))(x-2)^{10}} = 1 \), we will analyze the equation step by step.

### Step 1: Analyze the Absolute Value
The expression \( |x-2| \) can be split into two cases:
1. Case 1: \( x - 2 \geq 0 \) (i.e., \( x \geq 2 \))
2. Case 2: \( x - 2 < 0 \) (i.e., \( x < 2 \))

### Step 2: Case 1 - \( x \geq 2 \)
In this case, \( |x-2| = x-2 \). The equation becomes:
\[ (x-2)^{(\log_{3}(x^{4}) - 3\log_{x}(9))(x-2)^{10}} = 1 \]
For this equation to hold, the exponent must be zero or the base must be 1.

#### Subcase 1.1: Exponent is Zero
Set the exponent to zero:
\[ \log_{3}(x^{4}) - 3\log_{x}(9) + 10 = 0 \]
We can simplify \( \log_{x}(9) \) as follows:
\[ \log_{x}(9) = \frac{\log_{3}(9)}{\log_{3}(x)} = \frac{2}{\log_{3}(x)} \]
Thus, substituting into the equation gives:
\[ \log_{3}(x^{4}) - 3 \cdot \frac{2}{\log_{3}(x)} + 10 = 0 \]
This simplifies to:
\[ 4\log_{3}(x) - \frac{6}{\log_{3}(x)} + 10 = 0 \]
Let \( t = \log_{3}(x) \):
\[ 4t - \frac{6}{t} + 10 = 0 \]
Multiplying through by \( t \) (assuming \( t \neq 0 \)):
\[ 4t^2 + 10t - 6 = 0 \]

#### Subcase 1.2: Solve the Quadratic Equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 4 \cdot (-6)}}{2 \cdot 4} \]
\[ t = \frac{-10 \pm \sqrt{100 + 96}}{8} \]
\[ t = \frac{-10 \pm \sqrt{196}}{8} \]
\[ t = \frac{-10 \pm 14}{8} \]
Calculating the two possible values:
1. \( t = \frac{4}{8} = \frac{1}{2} \)
2. \( t = \frac{-24}{8} = -3 \)

#### Subcase 1.3: Convert Back to \( x \)
Recall \( t = \log_{3}(x) \):
1. If \( t = \frac{1}{2} \), then \( x = 3^{1/2} = \sqrt{3} \).
2. If \( t = -3 \), then \( x = 3^{-3} = \frac{1}{27} \).

However, both values \( \sqrt{3} \) and \( \frac{1}{27} \) are less than 2, which contradicts our assumption that \( x \geq 2 \).

### Step 3: Case 2 - \( x < 2 \)
In this case, \( |x-2| = 2-x \). The equation becomes:
\[ (2-x)^{(\log_{3}(x^{4}) - 3\log_{x}(9))(2-x)^{10}} = 1 \]
Again, for this equation to hold, the exponent must be zero or the base must be 1.

#### Subcase 2.1: Exponent is Zero
Set the exponent to zero:
\[ \log_{3}(x^{4}) - 3\log_{x}(9) + 10 = 0 \]
This is the same equation we derived earlier, so we can use the same \( t \) substitution:
\[ 4t + 10 - 6/t = 0 \]
This will yield the same solutions for \( t \).

### Step 4: Conclusion
Both cases yield the same values for \( x \) which do not satisfy the original conditions. Therefore, there are no values of \( x \) that satisfy the equation \( |x-2|^{(\log_{3}(x^{4}) - 3\log_{x}(9))(x-2)^{10}} = 1 \).