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A logarithmic graph changes quickly near its vertical asymptote and then more slowly as its input grows. This shape appears whenever an output records the logarithm of a positive ratio, including sound-level and pH scales. The base determines whether the graph increases or decreases, while transformations move the graph and its asymptote.
The following graph compares for several bases greater than . Every graph passes through and increases at a different rate.
For function with :
The following concrete examples show each step for drawing a logarithmic function graph.
Drawing
To draw this graph, we create a table of values by choosing values that are powers of :
Drawing
For base , the graph will be decreasing:
Place logarithmic graphs with different bases on one coordinate system to compare their shapes directly:
| Property | ||
|---|---|---|
| Graph direction | Increasing (monotonic) | Decreasing (monotonic) |
| Domain | ||
| Range | All real numbers | All real numbers |
| -intercept | ||
| Vertical asymptote |
Logarithmic graphs can be transformed in various ways:
We can shift the logarithmic function graph by adding or subtracting a constant to the function.
We can shift the logarithmic function graph by adding or subtracting a constant to the function.
Create a value table and draw the graphs of:
Determine the domain, range, and asymptote of function .
If and , determine:
Sketch the graph of and determine the -intercept.
Value tables:
For :
For :
Plot each ordered pair and draw a smooth curve that stays to the right of the vertical asymptote . The base graph rises, while the base graph falls.
For :
The logarithm is defined only when its argument is positive. The boundary of that condition gives the vertical asymptote:
For :
Replacing with shifts the graph right. The same expression must also remain positive:
For :
Adding outside the logarithm shifts every point vertically without changing the domain or the asymptote:
Published: . Updated: .
In particular, a -intercept would require , where the logarithm is undefined. The graph below confirms that it approaches but never reaches the -axis.
