# How do you graph ln(abs(x))?

Jan 24, 2016

The typical graph of just $\ln \left(x\right)$ is

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

Notice the domain restriction. In $\ln \left(x\right)$, $x > 0$. That is, negative numbers are not in the domain of a logarithmic function.

However, in $\ln \left(\left\mid x \right\mid\right)$, negative numbers are made positive.

For example, both ${e}^{2}$ and $- {e}^{2}$, when plugged into $\ln \left(\left\mid x \right\mid\right)$, result in $\ln \left({e}^{2}\right) = 2$.

In effect, adding the absolute value makes both the positive and negative realms available for the natural logarithm, in effect reflecting the graph over the $y$-axis, while retaining itself on the positive side: