College Algebra, Chapter 5, 5.2, Section 5.2, Problem 62

Graph the function $y = \ln |x|$, not by plotting points, but by starting from the graphs of family of Logarithmic functions. State the domain, range and asymptote.







By using the Properties of Absolute Value,

$\displaystyle y = \ln |x| \to y = \left\{ \begin{array}{cc}
\ln (x) & \text{for } x > 0 \\
\ln (-x) & \text{for } x < 0
\end{array} \right.$

So the graph of $y = \ln |x|$ is the graph of $y = \ln x$ for $x > 0$ and the graph of $y = \ln (x)$ that is reflected to the $y$-axis at $x > 0$. Its domain is all reals except and its range is all reals. The vertical asymptote is $x = 0$.