Let $f:,[2,\infty )\to \mathbb{R}$ be defined by $f(x)=ln(x-1)$. Determine whether $f$ is one‑to‑one on its domain and justify your answer.

Step-by-step solution

1. Find derivative: $f^{\prime }(x)=\frac{1}{x-1}$.
2. For $x>1$, $x-1>0$ ⇒ $f^{\prime }(x)>0$.
3. Thus $f$ is strictly increasing on $[2,infty)$.
4. A strictly increasing function is injective.
5. Therefore $f$ is one‑to‑one on its domain.

Concept tested: Monotonicity and injectivity

Published 15 September 2026 · How this solution was prepared and checked