Find the domain of
`log(x-2)-sqrt((3-x))`

Answer

To find the domain of the function \( f(x) = \log(x - 2) - \sqrt{3 - x} \), we need to consider the conditions for both the logarithmic and square root functions.

### Step 1: Determine the condition for the logarithmic function

The logarithmic function \( \log(x - 2) \) is defined when its argument is positive. Therefore, we need:

\[ x - 2 > 0 \]

This simplifies to:

\[ x > 2 \]

### Step 2: Determine the condition for the square root function

The square root function \( \sqrt{3 - x} \) is defined when its argument is non-negative. Therefore, we need:

\[ 3 - x \geq 0 \]

This simplifies to:

\[ x \leq 3 \]

### Step 3: Combine the conditions

Now we have two conditions:
1. \( x > 2 \)
2. \( x \leq 3 \)

To find the domain of the function, we need to find the intersection of these two conditions. This means we are looking for values of \( x \) that satisfy both conditions simultaneously.

### Step 4: Identify the intersection

The intersection of \( x > 2 \) and \( x \leq 3 \) is:

\[ 2 < x \leq 3 \]

### Conclusion

Thus, the domain of the function \( f(x) = \log(x - 2) - \sqrt{3 - x} \) is:

\[ \boxed{(2, 3]} \]

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