# What is the domain and range of ln(x-1)?

Mar 3, 2018

$x > 1$ (domain), $y \in \mathbb{R}$ (range)

#### Explanation:

The domain of a function is the set of all possible $x$ values that it is defined for, and the range is the set of all possible $y$ values. To make this more concrete, I'll rewrite this as:

$y = \ln \left(x - 1\right)$

Domain: The function $\ln x$ is defined only for all positive numbers. This means the value we're taking the natural log ($\ln$) of ($x - 1$) has to be greater than $0$.

Our inequality is as follows:

$x - 1 > 0$

Adding $1$ to both sides, we get:

$x > 1$ as our domain.

To understand the range, let's graph the function $y = \ln \left(x - 1\right)$.

graph{ln(x-1) [-10, 10, -5, 5]}

When we look at our graph, there are no discontinuities in it, thus our range is:

$y \in \mathbb{R}$, which just means $y$ is a member of the real numbers or $y$ can take on any value.