How to Make a Variation Table and Graph of ln(x) in TikZ

The natural logarithm, written $\ln x$, is defined only for $x>0$. Its variation table is short: the function increases from $-\infty$ to $+\infty$. The example below typesets that table and draws the graph using PGFPlots, a plotting package built on TikZ.

The code is a complete LaTeX document. Compile it with a current LaTeX distribution that includes PGFPlots.

Complete LaTeX example

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}

\[
\begin{array}{c|ccc}
x & 0^+ & & +\infty \\
\hline
\ln x & -\infty & \nearrow & +\infty
\end{array}
\]

\begin{center}
\begin{tikzpicture}
  \begin{axis}[
    width=10cm,
    height=7cm,
    axis lines=middle,
    xmin=0,
    xmax=4,
    ymin=-3,
    ymax=2,
    xlabel={$x$},
    ylabel={$y$},
    xtick={1,2,3,4},
    ytick={-3,-2,-1,0,1,2},
    grid=both,
    domain=0.08:4,
    samples=200,
    unbounded coords=jump
  ]
    \addplot[blue,thick] {ln(x)};
  \end{axis}
\end{tikzpicture}
\end{center}

\end{document}

The only package needed for the graph is pgfplots; it loads TikZ as a dependency. The compat setting selects PGFPlots’ compatibility behavior. If your installation is older and does not support version 1.18, change the setting to a version available in that installation.

Why the table has this form

The domain of $\ln x$ is $(0,\infty)$. On that domain,

$$\frac{d}{dx}\ln x=\frac{1}{x}>0.$$

A positive derivative means the function is strictly increasing everywhere it is defined. The endpoint behavior is

$$\lim_{x\to 0^+}\ln x=-\infty, \qquad \lim_{x\to +\infty}\ln x=+\infty.$$

In the array, the blank middle cell separates the two endpoint columns so the arrow can sit between them. The $0^+$ notation matters: the logarithm is not defined at zero, and the curve approaches the vertical axis without touching it.

How the graph settings work

The axis environment sets the visible window. Here, the horizontal range is from 0 to 4 and the vertical range is from $-3$ to 2. The curve itself is drawn only for $0.08\le x\le4$, because the logarithm cannot be evaluated at zero. The chosen lower plotting limit keeps the curve visible near the vertical axis; it does not change the function’s domain.

The expression {ln(x)} is PGFPlots’ natural-log function. With samples=200, PGFPlots evaluates it at many points and joins them to make a smooth curve. The curve passes through $(1,0)$ because $\ln 1=0$, and through $(e,1)$ because $\ln e=1$. The axis lines=middle option places the horizontal axis at $y=0$ and the vertical axis at $x=0$.

If you want to label the curve, add a node to the plot, for example:

\addplot[blue,thick] {ln(x)}
  node[pos=0.7,above] {$y=\ln x$};

Put this in place of the original \addplot line. The pos value controls where along the plotted curve the label appears.

If you mean log base a

For $y=\log_a x$, where $a>0$ and $a\ne1$, the domain remains $x>0$. If $a>1$, the function increases, and its limits at the endpoints are $-\infty$ and $+\infty$, just as for $\ln x$. If $0<a<1$, it decreases instead: it tends to $+\infty$ as $x\to0^+$ and to $-\infty$ as $x\to+\infty$.

PGFPlots’ ln(x) draws the natural logarithm. To plot a logarithm with another base, use the change-of-base formula $\log_a x=\ln x/\ln a$, so the plot expression becomes {ln(x)/ln(a)} with a numerical value substituted for a.

Common mistakes

• Plotting at $x=0$: The logarithm is undefined there. Start the plotted domain at a small positive number instead.
• Using a decreasing arrow for $\ln x$: Its derivative is positive on its entire domain, so the variation arrow points upward as $x$ increases.
• Confusing the plot window with the domain: A graph shown only up to $x=4$ is still a graph of a function defined for every $x>0$.
• Treating log and ln as interchangeable in code: In PGFPlots, use ln(x) for the natural logarithm. For a different base, apply the change-of-base formula.
• Expecting the curve to cross the vertical axis: The line $x=0$ is an asymptote, not part of the logarithm’s graph.