Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function . Because every logarithmic function of this form is the inverse of an exponential function with the form , their graphs will be reflections of each other across the line . To illustrate this, we can observe the relationship between the input and output values of and its equivalent in the table below.
x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
1 | 2 | 4 | 8 | ||||
–3 | –2 | –1 | 0 | 1 | 2 | 3 |
Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions and .
As we’d expect, the x– and y-coordinates are reversed for the inverse functions. The figure below shows the graph of f and g.

Figure 2. Notice that the graphs of and are reflections about the line y = x.
Observe the following from the graph:
- has a y-intercept at and has an x-intercept at .
- The domain of , , is the same as the range of .
- The range of , , is the same as the domain of .
A General Note: Characteristics of the Graph of the Parent Function, f(x) = logb(x)
For any real number x and constant b > 0, , we can see the following characteristics in the graph of :
- one-to-one function
- vertical asymptote: x = 0
- domain:
- range:
- x-intercept: and key point
- y-intercept: none
- increasing if
- decreasing if 0 < b < 1

Figure 3
Figure 3 shows how changing the base b in can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function has base

Figure 4. The graphs of three logarithmic functions with different bases, all greater than 1.
How To: Given a logarithmic function with the form , graph the function.
- Draw and label the vertical asymptote, x = 0.
- Plot the x-intercept, .
- Plot the key point .
- Draw a smooth curve through the points.
- State the domain, , the range, , and the vertical asymptote, x = 0.
Example 3: Graphing a Logarithmic Function with the Form .
Graph . State the domain, range, and asymptote.
Solution
Before graphing, identify the behavior and key points for the graph.
- Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote x = 0, and the right tail will increase slowly without bound.
- The x-intercept is .
- The key point is on the graph.
- We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

Figure 5. The domain is , the range is , and the vertical asymptote is x = 0.
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175. License: CC BY: Attribution. License Terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.