- Identify the domain of a logarithmic function
- Graph logarithmic functions
Domain of Logarithmic Functions
When working with logarithmic functions, understanding their domain is crucial. The domain of a function is the set of all possible input values ([latex]x[/latex]-values) that the function can accept without causing any mathematical issues. For logarithmic functions, this concept is especially important because it defines where the function is valid and where it isn’t.
Logarithmic functions are only defined for positive real numbers. This means you can only take the logarithm of a positive number.
For example, [latex]\mathrm{log}_{b}(x)[/latex] is only valid if [latex]x \gt 0[/latex]. Thus, the domain is [latex](0, \infty)[/latex]. This results in a vertical asymptote at [latex]x=0[/latex], which the graph approaches but never touches or crosses. The range of a logarithmic function is always all real numbers, [latex](-\infty, \infty)[/latex].
Recall that the exponential function is defined as [latex]y={b}^{x}[/latex] for any real number [latex]x[/latex] and constant [latex]b>0[/latex], [latex]b\ne 1[/latex], where
- 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].
In the last section we learned that the logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the inverse of the exponential function [latex]y={b}^{x}[/latex]. So, as inverse functions:
- 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].
Previously we saw that certain transformations can change the range of [latex]y={b}^{x}[/latex]. Similarly, applying transformations to the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] can change the domain. Therefore, when finding the domain of a logarithmic function, it is important to remember that the domain consists only of positive real numbers. That is, the value you are applying the logarithmic function to, also known as the argument of the logarithmic function, must be greater than zero.
[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].
- Set up an inequality showing the argument greater than zero.
- Solve for [latex]x[/latex].
- Write the domain in interval notation.