Solve for `x:log_(1//sqrt2),(x-1)gt2`

Answer

To solve the inequality \( \log_{\frac{1}{\sqrt{2}}}(x - 1) > 2 \), we will follow these steps:

### Step 1: Understand the logarithmic inequality
The base of the logarithm \( \frac{1}{\sqrt{2}} \) is less than 1. This means that the logarithmic function is decreasing. Therefore, the inequality \( \log_{\frac{1}{\sqrt{2}}}(x - 1) > 2 \) can be rewritten by flipping the inequality sign when we remove the logarithm.

### Step 2: Rewrite the inequality
Using the properties of logarithms, we can rewrite the inequality as:
\[ x - 1 < \left(\frac{1}{\sqrt{2}}\right)^2 \]

### Step 3: Calculate \( \left(\frac{1}{\sqrt{2}}\right)^2 \)
Calculating the right side:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
So the inequality now becomes:
\[ x - 1 < \frac{1}{2} \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ x < \frac{1}{2} + 1 \]
\[ x < \frac{3}{2} \]

### Step 5: Define the domain
Since we have \( \log_{\frac{1}{\sqrt{2}}}(x - 1) \), the argument \( x - 1 \) must be greater than 0:
\[ x - 1 > 0 \implies x > 1 \]

### Step 6: Combine the results
Now we have two conditions:
1. \( x > 1 \)
2. \( x < \frac{3}{2} \)

### Step 7: Intersection of the intervals
The solution set is the intersection of these two conditions:
\[ 1 < x < \frac{3}{2} \]

### Final Answer
Thus, the solution for \( x \) is:
\[ x \in (1, \frac{3}{2}) \]

---