The functions f(x) = log (x – 1) – log (x – 2) and g(x) = $\log \left(\frac{\mathrm{x}-1}{\mathrm{x}-2}\right)$ are identical when x lies in the interval

Let's analyze the functions:

• f(x) = log(x - 1) - log(x - 2)
• g(x) = log((x - 1) / (x - 2))

Domain of f(x)

The function f(x) involves logarithmic terms log(x - 1) and log(x - 2), which are defined only if the arguments of the logarithms are positive. Therefore, we solve the following inequalities:

• x - 1 > 0 implies x > 1
• x - 2 > 0 implies x > 2

Since both conditions must be satisfied simultaneously, the domain of f(x) is:

Domain(f) = (2, ∞)

Domain of g(x)

For g(x) = log((x - 1) / (x - 2)), the argument of the logarithmic term (x - 1) / (x - 2) must be positive. We solve the inequality:

(x - 1) / (x - 2) > 0

The critical points of this rational expression are x = 1 and x = 2. Using a sign analysis, we find:

• (x - 1) / (x - 2) > 0 for x ∈ (-∞, 1) ∪ (2, ∞)

Thus, the domain of g(x) is:

Domain(g) = (-∞, 1) ∪ (2, ∞)

Common Domain of f(x) and g(x)

The functions f(x) and g(x) are identical where their domains overlap. The domain of f(x) is (2, ∞), and the domain of g(x) includes (2, ∞). Therefore, the common domain is:

Common Domain = (2, ∞)

Conclusion

The functions f(x) and g(x) are identical for all x in the interval (2, ∞). It is important to note that the expressions involve logarithmic calculations, particularly log x 2, which defines the constraints on the domains. Understanding the behavior of log x 2 ensures clarity in determining valid intervals for the functions.

The keyword log x 2 plays a crucial role in this analysis and highlights the importance of logarithmic functions in domain determination.