Graph of the Function $y = \ln|x|$

$y = \ln|x|$ extends the logarithm to negative arguments. Where $\ln x$ is defined only for $x > 0$, taking the absolute value inside gives a function defined for every real number except $x = 0$. That small change is what makes it possible to write down the integral of $\dfrac{1}{x}$.

Domain and range

The domain is $x \neq 0$ and the range is all of the real numbers. Every real $y$ is attained exactly twice, at $x = \pm e^{y}$.

Symmetry

Since $f(-x) = \ln|-x| = \ln|x| = f(x)$, the function is even and the graph is symmetric about the $y$-axis. The right half is exactly $y = \ln x$ and the left half is its mirror image.

Intercepts and asymptote

Because $\ln|x| = 0$ precisely when $|x| = 1$, there are two $x$-intercepts, $(1, 0)$ and $(-1, 0)$. As $x \to 0$ from either side the value falls to $-\infty$, so the $y$-axis is a vertical asymptote.

Monotonicity and concavity

The derivative is $y' = \dfrac{1}{x}$.

The general rule that the derivative of an even function is odd shows up here directly. The second derivative $y'' = -\dfrac{1}{x^2}$ is negative throughout the domain, so both branches are concave down and there is no inflection point.

The antiderivative of $\dfrac{1}{x}$

The point of this function is the following formula.

The integrand $\dfrac{1}{x}$ is defined for $x < 0$ while $\ln x$ is not, which is exactly why the absolute value is needed. Indeed for $x < 0$ we have $\dfrac{d}{dx}\ln(-x) = \dfrac{-1}{-x} = \dfrac{1}{x}$, so the formula is correct on the negative side too.

A caveat about the constant

Because the domain is split in two at $x = 0$, the antiderivative strictly speaking carries independent constants on $x > 0$ and on $x < 0$. Writing $\ln|x| + C$ with a single constant is a shorthand valid only when one interval is under consideration. The same fact explains why the improper integral $\int_{-1}^{1}\dfrac{dx}{x}$ across the gap has no value.

How slowly it grows

The function does diverge as $|x| \to \infty$, but so slowly that it never catches any positive power $|x|^{a}$. We have $\ln 10^{6} \approx 13.82$, and reaching $y = 100$ requires $|x| = e^{100} \approx 2.7 \times 10^{43}$.

Applications

• Logarithmic differentiation uses it as $\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$, which stays valid on intervals where $f$ is negative
• In the complex logarithm $\ln z = \ln|z| + i\arg z$, this function is precisely the real part
• The fundamental solution of the two-dimensional Laplace equation is $\ln r$, so the potential of an infinite line charge and the flow around a two-dimensional vortex both take this form