Find for what values of `x` the following functions would be identical.
`f(x)=log(x-1)-log(x-2)` and `g(x)=log((x-1)/(x-2))`

Explanation

To determine the values of \( x \) for which the functions \( f(x) = \log(x-1) - \log(x-2) \) and \( g(x) = \log\left(\frac{x-1}{x-2}\right) \) are identical, we can follow these steps:

### Step 1: Rewrite \( f(x) \)
Using the properties of logarithms, we can rewrite \( f(x) \):
\[ f(x) = \log(x-1) - \log(x-2) = \log\left(\frac{x-1}{x-2}\right) \]

### Step 2: Set the functions equal
Now that we have rewritten \( f(x) \), we can set the two functions equal to each other:
\[ \log\left(\frac{x-1}{x-2}\right) = \log\left(\frac{x-1}{x-2}\right) \]
This shows that the two functions are equal for all \( x \) in their domain, provided that the arguments of the logarithms are positive.

### Step 3: Determine the domain of \( f(x) \)
For \( f(x) \) to be defined, the arguments of the logarithms must be positive:
1. \( x - 1 > 0 \) implies \( x > 1 \)
2. \( x - 2 > 0 \) implies \( x > 2 \)

The stricter condition is \( x > 2 \). Thus, the domain of \( f(x) \) is:
\[ x \in (2, \infty) \]

### Step 4: Determine the domain of \( g(x) \)
For \( g(x) \) to be defined, the argument of the logarithm must also be positive:
\[ \frac{x-1}{x-2} > 0 \]
This inequality holds when both the numerator and denominator are either both positive or both negative.

1. **Both positive**:
- \( x - 1 > 0 \) implies \( x > 1 \)
- \( x - 2 > 0 \) implies \( x > 2 \)

Thus, for both to be positive, \( x > 2 \).

2. **Both negative**:
- \( x - 1 < 0 \) implies \( x < 1 \)
- \( x - 2 < 0 \) implies \( x < 2 \)

Thus, for both to be negative, \( x < 1 \).

Combining these, the domain of \( g(x) \) is:
\[ x \in (-\infty, 1) \cup (2, \infty) \]

### Step 5: Find the intersection of the domains
The domains of \( f(x) \) and \( g(x) \) are:
- \( f(x): (2, \infty) \)
- \( g(x): (-\infty, 1) \cup (2, \infty) \)

The intersection of these two domains is:
\[ (2, \infty) \]

### Conclusion
Thus, the values of \( x \) for which the functions \( f(x) \) and \( g(x) \) are identical is:
\[ x \in (2, \infty) \]