Logarithm
2026-02-28 17:13 Diff
https://brightchamps.com/en-us/math/algebra/logarithm

Finding points that describe the behavior of a function and drawing a curve through those points is known as graphing a logarithmic function. Depending on the base value, the curve can either increase or decrease.

The curve increases if the base is greater than 1 \( (base>1)\), and decreases if the base lies between 0 and 1 \((0 < \text{base} < 1)\). To graph a logarithmic function, we must follow certain steps:

Step 1: Identify the domain and range.

Step 2: Find the vertical asymptote by setting the argument equal to 0. Remember that a logarithmic graph has a vertical asymptote but no horizontal asymptote.

Step 3: Set the argument equal to 1 by substituting the value of x. To find the x-intercept, use the property log⁡a(1) = 0.

Step 4: Set the argument equal to the base by substituting the value of x. To find another point on the graph, use the property log⁡a(a) = 1.

Step 5: Draw the curve by connecting the two points and extending the curve toward the vertical asymptote.

Let’s look at an example to make it easier to understand.


Consider the logarithmic function:

\(f(x) = \log_{2}(x - 1)\)

Step 1: The basic form of the logarithmic function is \(f(x) = \log_a(x).\)


Here, the base is a=2. Since 2>1, the curve will increase.

Step 2: Now set the argument greater than 0:

\(x - 1 > 0 \quad \Rightarrow \quad x > 1\)

So, the domain is (1,∞).


\(Range = R.\)

Step 3: Find the vertical asymptote by setting the argument equal to 0. In a logarithmic function, the argument must be positive (> 0).

\(x−1=0⇒x=1\)

So, the vertical asymptote is at x=1.

Step 4: Next, let’s find some points.

At x = 2:

\(f(2) = \log_{2}(2 - 1) = \log_{2}(1) = 0\)

So, the point is (2,0).

At x = 3:

\(f(3) = \log_{2}(3 - 1) = \log_{2}(2) = 1\)

So, the point is (3,1).

Step 5: Draw the graph by connecting the points, starting near the asymptote x=1.

Here, the curve passes through the points (2,0) and (3, 1), and increases gradually. The red line shows the vertical asymptote at x=1.