Chapter 6 Exponential and Logarithmic Functions
6.4 Graphs of Logarithmic Functions
Learning Objectives
In this section, you will:
- Identify the domain of a logarithmic function.
- Graph logarithmic functions.
- Graph transformations of logarithmic functions.
In Section 6.2, Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.
To illustrate, suppose we invest[latex]\,\text{\$}2500\,[/latex]in an account that offers an annual interest rate of[latex]\,5[/latex]% compounded continuously. We already know that the balance in our account for any year[latex]\,t\,[/latex]can be found with the equation[latex]\,A=2500{e}^{0.05t}.[/latex]
But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.
In this section, we will discuss the values for which a logarithmic function is defined and then turn our attention to graphing the family of logarithmic functions.
Identify the Domain of a Logarithmic Function
Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.
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]
Transformations of the parent function[latex]\,y={\mathrm{log}}_{b}\left(x\right)\,[/latex]behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, an