Characteristics of Graphs of Logarithmic Functions
Learning Outcomes
- Determine the domain and range of a logarithmic function.
- Determine the x-intercept and vertical asymptote of a logarithmic function.
- Identify whether a logarithmic function is increasing or decreasing and give the interval.
- Identify the features of a logarithmic function that make it an inverse of an exponential function.
- The domain of [latex]y[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].
- The range of [latex]y[/latex] is [latex]\left(0,\infty \right)[/latex].
- The domain of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the range of [latex]y={b}^{x}[/latex]: [latex]\left(0,\infty \right)[/latex].
- The range of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the domain of [latex]y={b}^{x}[/latex]: [latex]\left(-\infty ,\infty \right)[/latex].
[latex]\begin{array}{l}2x - 3>0\hfill & \text{Show the argument greater than zero}.\hfill \\ 2x>3\hfill & \text{Add 3}.\hfill \\ x>\dfrac{3}{2}\hfill & \text{Divide by 2}.\hfill \end{array}[/latex]
In interval notation, the domain of [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)[/latex] is [latex]\left(\dfrac{3}{2},\infty \right)[/latex].Tip for success
Finding the domain of the logarithm function is similar to finding the domain of the square root function. Recall that for [latex]y = \sqrt{u}, u\geq0[/latex], so the domain of [latex]y=\sqrt{2x-3}[/latex] is all [latex]x[/latex] for which [latex]2x-3\geq0[/latex], or all [latex]x\geq\dfrac{3}{2}[/latex]. Since the logarithm function takes input only strictly greater than zero, we see in the example above that the domain of [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)[/latex] is [latex]\left(\dfrac{3}{2},\infty \right)[/latex].How To: Given a logarithmic function, identify the domain
- Set up an inequality showing the argument greater than zero.
- Solve for [latex]x[/latex].
- Write the domain in interval notation.
Example: Identifying the Domain Resulting From a Logarithmic Shift
What is the domain of [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)[/latex]?Answer:
[latex]\begin{array}{l}x+3>0\hfill & \text{The argument must be positive}.\hfill \\ x>-3\hfill & \text{Subtract 3}.\hfill \end{array}[/latex]
The logarithmic function is defined only when the argument is positive, so this function is defined when [latex]x+3>0[/latex]. The domain of [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)[/latex] is [latex]\left(-3,\infty \right)[/latex].Try It
What is the domain of [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x - 2\right)+1[/latex]?Answer: [latex]\left(2,\infty \right)[/latex]
Example: Identifying the Domain Resulting From a Logarithmic Shift and Reflection
What is the domain of [latex]f\left(x\right)=\mathrm{log}\left(5 - 2x\right)[/latex]?Answer: The logarithmic function is defined only when the argument is positive, so this function is defined when [latex]5 - 2x>0[/latex]. Solving this inequality,
[latex]\begin{array}{l}5 - 2x>0\hfill & \text{The argument must be positive}.\hfill \\ -2x>-5\hfill & \text{Subtract }5.\hfill \\ x<\frac{5}{2}\hfill & \text{Divide by }-2\text{ and switch the inequality}.\hfill \end{array}[/latex]
The domain of [latex]f\left(x\right)=\mathrm{log}\left(5 - 2x\right)[/latex] is [latex]\left(-\infty ,\frac{5}{2}\right)[/latex].Try It
What is the domain of [latex]f\left(x\right)=\mathrm{log}\left(x - 5\right)+2[/latex]?Answer: [latex]\left(5,\infty \right)[/latex]
[ohm_question]14399[/ohm_question]Graphing a Logarithmic Function Using a Table of Values
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 [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all of its transformations: shifts, stretches, compressions, and reflections. We begin with the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. Because every logarithmic function of this form is the inverse of an exponential function of the form [latex]y={b}^{x}[/latex], their graphs will be reflections of each other across the line [latex]y=x[/latex]. To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[/latex] and its equivalent logarithmic form [latex]x={\mathrm{log}}_{2}\left(y\right)[/latex] in the table below.recall inverse relationships
Recall that if an invertible function [latex]f(x)[/latex] contains a point, [latex]\left(a, b\right)[/latex], then the inverse function [latex]f^{-1}\left(x\right)[/latex] must contain the point [latex]\left(b, a\right)[/latex].x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
[latex]{2}^{x}=y[/latex] | [latex]\frac{1}{8}[/latex] | [latex]\frac{1}{4}[/latex] | [latex]\frac{1}{2}[/latex] | 1 | 2 | 4 | 8 |
[latex]{\mathrm{log}}_{2}\left(y\right)=x[/latex] | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
[latex]f\left(x\right)={2}^{x}[/latex] | [latex]\left(-3,\frac{1}{8}\right)[/latex] | [latex]\left(-2,\frac{1}{4}\right)[/latex] | [latex]\left(-1,\frac{1}{2}\right)[/latex] | [latex]\left(0,1\right)[/latex] | [latex]\left(1,2\right)[/latex] | [latex]\left(2,4\right)[/latex] | [latex]\left(3,8\right)[/latex] |
[latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex] | [latex]\left(\frac{1}{8},-3\right)[/latex] | [latex]\left(\frac{1}{4},-2\right)[/latex] | [latex]\left(\frac{1}{2},-1\right)[/latex] | [latex]\left(1,0\right)[/latex] | [latex]\left(2,1\right)[/latex] | [latex]\left(4,2\right)[/latex] | [latex]\left(8,3\right)[/latex] |
- [latex]f\left(x\right)={2}^{x}[/latex] has a y-intercept at [latex]\left(0,1\right)[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex] has an x-intercept at [latex]\left(1,0\right)[/latex].
- The domain of [latex]f\left(x\right)={2}^{x}[/latex], [latex]\left(-\infty ,\infty \right)[/latex], is the same as the range of [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].
- The range of [latex]f\left(x\right)={2}^{x}[/latex], [latex]\left(0,\infty \right)[/latex], is the same as the domain of [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].
A General Note: Characteristics of the Graph of the Parent Function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]
For any real number x and constant b > 0, [latex]b\ne 1[/latex], we can see the following characteristics in the graph of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]:- one-to-one function
- vertical asymptote: x = 0
- domain: [latex]\left(0,\infty \right)[/latex]
- range: [latex]\left(-\infty ,\infty \right)[/latex]
- x-intercept: [latex]\left(1,0\right)[/latex] and key point [latex]\left(b,1\right)[/latex]
- y-intercept: none
- increasing if [latex]b>1[/latex]
- decreasing if 0 < b < 1
Try it
Use an online graphing tool to graph the function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] Adjust the [latex]b[/latex] value to investigate how changes in the base of the logarithmic function affect the graph of that function. Which function is an increasing function? Which is decreasing? Does the x-intercept change when you change the base? Make sure to use values between [latex]0[/latex] and [latex]1[/latex] as well as values greater than [latex]1[/latex].How To: Given a logarithmic function Of the form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex], graph the function
- Draw and label the vertical asymptote, x = 0.
- Plot the x-intercept, [latex]\left(1,0\right)[/latex].
- Plot the key point [latex]\left(b,1\right)[/latex].
- Draw a smooth curve through the points.
- State the domain, [latex]\left(0,\infty \right)[/latex], the range, [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote, x = 0.
Example: Graphing a Logarithmic Function Of the Form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]
Graph [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex]. State the domain, range, and asymptote.Answer: 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 [latex]\left(1,0\right)[/latex].
- The key point [latex]\left(5,1\right)[/latex] is on the graph.
- We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Try It
Graph [latex]f\left(x\right)={\mathrm{log}}_{\frac{1}{5}}\left(x\right)[/latex]. State the domain, range, and asymptote.Answer: The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.
[ohm_question]34999[/ohm_question] [ohm_question]35000[/ohm_question]Licenses & Attributions
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- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
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CC licensed content, Shared previously
- Questoin ID 34999, 35000. Authored by: Smart, JIm. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].