4.5: Graphs and Properties of Logarithmic Functions
In this section, you will:
- examine properties of logarithmic functions
- examine graphs of logarithmic functions
- examine the relationship between graphs of exponential and logarithmic functions
Recall that the exponential function \(f(x) = 2^x\) produces this table of values
| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| \(f(x)\) | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 |
Since the logarithmic function is an inverse of the exponential, \(g(x)=\log_{2}(x)\) produces the table of values
| \(x\) | 1/8 | 1/4 | 1/2 | 1 | 2 | 4 | 8 |
| \(g(x)\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
In this second table, notice that
- As the input increases, the output increases.
- As input increases, the output increases more slowly.
- Since the exponential function only outputs positive values, the logarithm can only accept positive values as inputs, so the domain of the log function is \((0, \infty)\).
- Since the exponential function can accept all real numbers as inputs, the logarithm can have any real number as output, so the range is all real numbers or \((-\infty, \infty)\).
Plotting the graph of \(g(x) = \log_{2}(x)\) from the points in the table , notice that as the input values for \(x\) approach zero, the output of the function grows very large in the negative direction, indicating a vertical asymptote at \(x = 0\).
In symbolic notation we write
as \(x \rightarrow 0^{+}\), \(f(x) \rightarrow-\infty\)
and as \(x \rightarrow \infty, f(x) \rightarrow \infty\)
Source: The material in this section of the textbook originates from David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, “ Chapter 4: Exponential and Logarithmic Functions ,” licensed under a Creative Commons CC BY-SA 3.0 license. The material here is based on material contained in that textbook but has been modified by Roberta Bloom, as permitted under this license.
Graphically, in the function \(g(x) = \log_{b}(x)\), \(b > 1\), we observe the following properties:
- The graph has a horizontal intercept at (1, 0)
- The line x = 0 (the y-axis) is a vertical asymptote; as \(x \rightarrow 0^{+}, y \rightarrow-\infty\)
- The graph is increasing if \(b > 1\)
- The domain of the function is \(x > 0\), or (0, \(\infty\))
- The range of the function is all real numbers, or \((-\infty, \infty)\)
However if the base \(b\) is less than 1, 0 < \(b\) < 1, then the graph appears as below.
This follows from the log property of reciprocal bases : \(\log _{1 / b} C=-\log _{b}(C)\)
- The graph has a horizontal intercept at (1, 0)
- The line x = 0 (the y-axis) is a vertical asymptote; as \(x \rightarrow 0^{+}, y \rightarrow \infty\)
- The graph is decreasing if 0 < \(b\) < 1
- The domain of the function is \(x\) > 0, or (0, \(\infty\))
- The range of the function is all real numbers, or \((-\infty, \infty)\)
When graphing a logarithmic function, it can be helpful to remember that the graph will pass through the points (1, 0) and (\(b\), 1).
Finally, we compare the graphs of \(y = b^x\) and \(y = \log_{b}(x)\), shown below on the same axes.
Because the functions are inverse functions of each other, for every specific ordered pair
(\(h\), \(k\)) on the graph of \(y = b^x\), we find the point (\(k\), \(h\)) with the coordinates reversed on the graph of \(y = \log_{b}(x)\).
In other words, if the point with \(x = h\) and \(y = k\) is on the graph of \(y = b^x\), then the point with \(x = k\) and \(y = h\) lies on the graph of \(y = \log_{b} (x)\)
The domain of \(y = b^x\) is the range of \(y = \log_{b} (x)\)
The range of \(y = b^x\) is the domain of \(y = \log_{b} (x)\)
For this reason, the graphs appear as reflections, or mirror images, of each other across the diagonal line \(y=x\). This is a property of graphs of inverse functions that students should recall from their study of inverse functions in their prerequisite algebra class.
| \(\bf{y = b^x}\) , with \(\bf{b>1}\) | \(\bf{y = \log_{b} (x)}\) , with \(\bf{b>1}\) | |
|---|---|---|
| Domain |
all real numbers |
all positive real numbers |
| Range |
all positive real numbers |
all real numbers |
| Intercepts |
(0,1) |
(1,0) |
| Asymptotes |
Horizontal asymptote is
As \(x \rightarrow-\infty, \: y \rightarrow 0\) |
Vertical asymptote is
As \(x \rightarrow 0^{+}, \: y \rightarrow-\infty\) |
Product Property: \(y = \log_{b} (ac)\) = \(\log_{b} (a)\) + \(\log_{b} (c)\)
Quotient Property: \(y = \log_{b} \frac{a}{c}\) = \(\log_{b} (a)\) - \(\log_{b} (c)\)
Power Property: \(y = \log_{b} a^c\) = c\(\log_{b} (a)\)
Source: The material in this section of the textbook originates from David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, “ Chapter 4: Exponential and Logarithmic Functions ,” licensed under a Creative Commons CC BY-SA 3.0 license. The material here is based on material contained in that textbook but has been modified by Roberta Bloom, as permitted under this license.