If the function `f(x)=log(x-2)-log(x-3) and g(x)=log((x-2)/(x-3))` are identical, then

Options

A. `x in [2,3]`
B. `x in [2,oo)`
C. ` x in (3,oo)`
D. `x in R `

Answer

To determine if the functions \( f(x) = \log(x-2) - \log(x-3) \) and \( g(x) = \log\left(\frac{x-2}{x-3}\right) \) are identical, we need to follow these steps:

### Step 1: Find the domain of \( f(x) \)

The function \( f(x) \) involves logarithmic expressions. The logarithm is defined only for positive arguments. Therefore, we need:

1. \( x - 2 > 0 \) → \( x > 2 \)
2. \( x - 3 > 0 \) → \( x > 3 \)

The more restrictive condition is \( x > 3 \). Thus, the domain of \( f(x) \) is:

\[ \text{Domain of } f(x) = (3, \infty) \]

### Step 2: Find the domain of \( g(x) \)

For \( g(x) \), we also have a logarithmic function, and we need the argument to be positive:

\[ \frac{x-2}{x-3} > 0 \]

This inequality holds when both the numerator and denominator are either both positive or both negative.

- **Case 1**: Both positive:
- \( x - 2 > 0 \) → \( x > 2 \)
- \( x - 3 > 0 \) → \( x > 3 \)

This leads to \( x > 3 \).

- **Case 2**: Both negative:
- \( x - 2 < 0 \) → \( x < 2 \)
- \( x - 3 < 0 \) → \( x < 3 \)

This leads to \( x < 2 \).

Combining these cases, we find that \( g(x) \) is defined for:

\[ \text{Domain of } g(x) = (-\infty, 2) \cup (3, \infty) \]

### Step 3: Determine the common domain

The common domain for both functions is where they overlap. Since \( f(x) \) is defined for \( (3, \infty) \) and \( g(x) \) is defined for \( (3, \infty) \) as well, the common domain is:

\[ \text{Common Domain} = (3, \infty) \]

### Step 4: Verify if \( f(x) = g(x) \) in the common domain

Using the logarithmic identity:

\[ \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \]

We can rewrite \( f(x) \):

\[ f(x) = \log(x-2) - \log(x-3) = \log\left(\frac{x-2}{x-3}\right) = g(x) \]

Thus, \( f(x) = g(x) \) for all \( x \) in the common domain \( (3, \infty) \).

### Conclusion

Since both functions have the same domain and are equal within that domain, we conclude that the functions \( f(x) \) and \( g(x) \) are identical in the interval:

\[ \text{The functions are identical for } x \in (3, \infty) \]

### Final Answer

The correct option is \( \text{Option C: } x \text{ belongs to } (3, \infty) \).

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