If `log_(2)(x-2)ltlog_(4)(x-2)`,, find the interval in which x lies.

Answer

To solve the inequality \( \log_{2}(x-2) < \log_{4}(x-2) \), we will follow these steps:

### Step 1: Rewrite the logarithm
We know that \( \log_{4}(x-2) \) can be rewritten in terms of base 2:
\[ \log_{4}(x-2) = \frac{\log_{2}(x-2)}{\log_{2}(4)} = \frac{\log_{2}(x-2)}{2} \]
So the inequality becomes:
\[ \log_{2}(x-2) < \frac{1}{2} \log_{2}(x-2) \]

### Step 2: Rearrange the inequality
Now, we can rearrange the inequality:
\[ \log_{2}(x-2) - \frac{1}{2} \log_{2}(x-2) < 0 \]
This simplifies to:
\[ \frac{1}{2} \log_{2}(x-2) < 0 \]

### Step 3: Multiply by 2
To eliminate the fraction, we can multiply both sides by 2 (note that multiplying by a positive number does not change the direction of the inequality):
\[ \log_{2}(x-2) < 0 \]

### Step 4: Exponentiate to eliminate the logarithm
Now, we exponentiate both sides to solve for \( x \):
\[ x-2 < 2^0 \]
Since \( 2^0 = 1 \), we have:
\[ x - 2 < 1 \]

### Step 5: Solve for \( x \)
Adding 2 to both sides gives:
\[ x < 3 \]

### Step 6: Consider the domain
Since we are dealing with logarithms, we must also consider the domain of the logarithmic function. The expression \( x-2 \) must be greater than 0:
\[ x - 2 > 0 \implies x > 2 \]

### Final Solution
Combining both conditions, we find:
\[ 2 < x < 3 \]

### Summary
The interval in which \( x \) lies is \( (2, 3) \).