# Exponential and Logarithmic Functions

## Graphs of Logarithmic Functions


### Learning Objectives

1. Find the domain and range of a relation and a function. (IA 3.5.1)
2. Graph Logarithmic functions. (IA 10.3.3)

### Objective 1: Find the domain and range of a relation and a function. (IA 3.5.1)


### Practice Makes Perfect 


Find the domain and range of a relation and a function.


### Objective 2: Graph Logarithmic functions. (IA 10.3.3)

To graph a logarithmic function  , it is easiest to convert the equation to its exponential form,  . Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.

### Practice Makes Perfect 


Graph Logarithmic functions
In 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 
 in an account that offers an annual interest rate of 
 compounded continuously. We already know that the balance in our account for any year 
 can be found with the equation 
 
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?  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.

### Finding 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 
 for any real number 
 and constant 
 
 where
1. The domain of 
 is 
 
2. The range of 
 is 
 
In the last section we learned that the logarithmic function 
 is the inverse of the exponential function 
 So, as inverse functions:
1. The domain of 
 is the range of 
 
 
2. The range of 
 is the domain of 
 
 
Transformations of the parent function 
 behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections.
In Graphs of Exponential Functions we saw that certain transformations can change the range of 
 Similarly, applying transformations to the parent function 
 can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.
For example, consider 
 This function is defined for any values of 
 such that the argument, in this case 
 is greater than zero. To find the domain, we set up an inequality and solve for 
 
In interval notation, the domain of 
 is 
 

### Graphing Logarithmic Functions

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 
 along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function 
 Because every logarithmic function of this form is the inverse of an exponential function with the form 
 their graphs will be reflections of each other across the line 
 To illustrate this, we can observe the relationship between the input and output values of 
 and its equivalent 
 in .
Using the inputs and outputs from  , we can build another table to observe the relationship between points on the graphs of the inverse functions 
 and 
 See .
As we’d expect, the x- and y-coordinates are reversed for the inverse functions.  shows the graph of 
 and 
 
Observe the following from the graph:
1.  
 has a y-intercept at 
 and 
 has an x- intercept at 
 
2. The domain of 
 
 is the same as the range of 
 
3. The range of 
 
 is the same as the domain of 
 

### Graphing Transformations of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function 
 without loss of shape.

### Graphing a Horizontal Shift of f(x) = log(x)

When a constant 
 is added to the input of the parent function 
 the result is a horizontal shift 
 units in the opposite direction of the sign on 
 To visualize horizontal shifts, we can observe the general graph of the parent function 
 and for 
 alongside the shift left, 
 and the shift right, 
 See .

### Graphing a Vertical Shift of y = log(x)

When a constant 
 is added to the parent function 
 the result is a vertical shift 
 units in the direction of the sign on 
 To visualize vertical shifts, we can observe the general graph of the parent function 
 alongside the shift up, 
 and the shift down, 
 See .

### Graphing Stretches and Compressions of y = log(x)

When the parent function 
 is multiplied by a constant 
 the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set 
 and observe the general graph of the parent function 
 alongside the vertical stretch, 
 and the vertical compression, 
 See .

### Graphing Reflections of f(x) = log(x)

When the parent function 
 is multiplied by 
 the result is a reflection about the x-axis. When the input is multiplied by 
 the result is a reflection about the y-axis. To visualize reflections, we restrict 
 and observe the general graph of the parent function 
 alongside the reflection about the x-axis, 
 and the reflection about the y-axis, 
 

### Summarizing Translations of the Logarithmic Function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in  to arrive at the general equation for translating exponential functions.

### Key Equations


### Key Concepts

1. To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for 
 See  and 
2. The graph of the parent function 
 has an x-intercept at 
 domain 
 range 
 vertical asymptote 
 and


See .

3. The equation 
 shifts the parent function 
 horizontally


See .
4. The equation 
 shifts the parent function 
 vertically

See .
5. For any constant 
 the equation 



See  and .

6. When the parent function 
 is multiplied by 
 the result is a reflection about the x-axis. When the input is multiplied by 
 the result is a reflection about the y-axis.


See .


7. All translations of the logarithmic function can be summarized by the general equation 
 See .
8. Given an equation with the general form 
 we can identify the vertical asymptote 
 for the transformation. See .
9. Using the general equation 
 we can write the equation of a logarithmic function given its graph. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, state the domain and range of the function.
For the following exercises, state the domain and the vertical asymptote of the function.
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

### Graphical

For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, sketch the graphs of each pair of functions on the same axis.
For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, sketch the graph of the indicated function.
For the following exercises, write a logarithmic equation corresponding to the graph shown.

### Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

### Extensions
