How Do You Graph a Logarithmic Function?

Graph a logarithmic function by identifying its vertical asymptote, plotting several convenient points, and drawing a smooth curve through them. For the basic function $y=\log_a x$, where $a>0$ and $a\neq1$, the asymptote is $x=0$, the domain is $x>0$, and the graph passes through $(1,0)$.

The Reasoning

A logarithmic function is the inverse of an exponential function:

$$y=\log_a x \quad\Longleftrightarrow\quad x=a^y.$$

Therefore, the graph of $y=\log_a x$ is the reflection of $y=a^x$ in the line $$. For example, the exponential points , , and become , , and .

For example, to graph $y=\log_2 x$, plot

$$\left(\frac{1}{2},-1\right),\quad (1,0),\quad (2,1),\quad (4,2).$$

Then draw a smooth increasing curve approaching, but never touching, $x=0$.

For a transformed function such as $y=\log_a(x-h)+k$, the vertical asymptote becomes $x=h$, and the domain becomes . Translate the basic graph units horizontally and units vertically.

Use this reliable sketching process:

1. Set the logarithm's argument greater than zero to find the domain. Its boundary gives the vertical asymptote.
2. Find the x-intercept by setting $y=0$; since $\log_a 1=0$, set the argument equal to $1$.
3. Find the y-intercept, if it exists, by substituting .

The range remains all real numbers in both cases.

A common misconception is that the logarithmic graph crosses its vertical asymptote. It does not: the logarithm is undefined when its argument is zero or negative.

Exam Technique

This is IB Mathematics AA SL 2.9 and may appear on Paper 1 or Paper 2. In a sketch, examiners expect a labelled asymptote, correct intercepts, the correct increasing or decreasing shape, and an indication of the domain; do not rely only on a GDC plot.