11th Grade Q1

To determine the asymptote of a logarithmic function, you identify the vertical line that the graph approaches but never crosses.

A logarithmic function generally has the form:

where:

• $a$ is a constant multiplier,
• $b$ is the base of the logarithm,
• $x+d$ is the argument of the logarithm,
• $c$ is a constant shift.

Method

1. Determine the argument of the logarithm

The argument inside the logarithm must satisfy:

Solve this inequality for $x$ to find the domain of the function.

2. Find the value where the argument equals zero

The logarithmic function approaches $-\infty$ as the argument approaches $0$ from the positive side. This defines the vertical asymptote:

Thus, the vertical asymptote is the line:

Example

Find the asymptote of $f(x)=3\cdot {\log }_2(x-1)+4$.

1. Determine the argument of the logarithm:
   The argument is $x-1$. For the function to be defined

So the domain is $x>1$.

1. Find the vertical asymptote:
   The vertical asymptote occurs where the argument equals $0$

Thus, the vertical asymptote is the line

The asymptote is usually drawn as a dotted line. Here is the graph of $f(x)=3\cdot {\log }_2(x-1)+4$

Key Points to Remember

• The vertical asymptote occurs where the argument of the logarithm equals $0$.
• The vertical asymptote is always a vertical line written as $x=\text{value}$.