The graph of a basic logarithm is relatively simple. This lesson will show you how to graph a logarithm and what the transformations will do to the graph as well as their effects on the domain and range of the graph.
What Is a Logarithm?
The basic formula for a logarithm (log) is y = log2x is equivalent to 2y = x which means that the solution to a logarithm equation is the power you must raise a certain number to in order to obtain another number. The logarithm function is the inverse of an exponential, which is a term that has a variable in its exponent. For example, 2^x is an exponential.
It stands to reason, then, that the graph of a logarithm would be the inverse of the graph of an exponential. And you can see from this picture below that that is correct. The graph of the logarithm is just the graph of the exponential inverted across a straight line.
Graph of logarithm and exponential
How to Graph a Logarithm
There are two methods you can use in order to graph a logarithm.
The first is to substitute in numbers for one variable to get values for the other, then plot them on a graph and connect the dots. When using this method, remember that the log is undefined at zero and less than zero, so x can only be greater than zero.
Let's graph the following as an example: log2x = y
The first step is to draw a chart, then fill in the values for x and y.
You can pick any value for y that you desire, but smaller is usually better. This way your graph is not huge. You can get the general idea for the graph from 5 or so points.
Then, when you have your points, just plot them on the graph and connect the dots. Remember, with the graph of a general logarithm, it will never touch or cross the y-axis but will come as close as possible.
Graph using first method
The second method for graphing a logarithm is to use a graphing calculator. Read the instructions that came with your calculator in order to graph logarithms using this method.
The Domain and Range of Logarithms
The domain of any equation is the possible x values for that equation. It is any number that x could possibly be when that equation is graphed on a coordinate plane. For the general logarithm equation, the domain is x > 0 because x cannot be zero or below.
The range, or possible y values for any equation, is any number that y could possibly be when the equation is graphed on the coordinate plane. For the general logarithm equation, the range is y = all real numbers.
Graphing Transformations of the Logarithm Equation
So far, we have only talked about the general logarithm equation and its graph, but what if the equation is more complex, such as y = log2x + 2
Adding or subtracting to the equation causes the graph to shift, either up, down, left or right, depending on how the addition or subtraction is framed. Graphing logarithmic equations that have been shifted can be done very easily if you remember the set of rules that govern these shifts.
Rules for Graphing Transformations:
1. The first rule says that adding a number to the equation will cause the graph to shift up the number of spaces indicated by that number. The example shown previously has a +2 added to the equation, which means that the graph will shift up two spaces from the general graph.
This graph is shifted up.
As you can see above, the graph has been shifted up two spaces.
2. The second rule states that subtracting a number from the equation will cause the graph to shift down that number of spaces. Take a look at this example: y = log2x - 2
Because of the -2 in the equation, the graph will be shifted down two spaces.
This graph is shifted down.
3. The third rule says that adding a number inside the logarithmic argument will cause the graph to shift left. This time, the number being added will be in parentheses with the x, indicating that it is a part of the log function and not just a number to be added on at the end of the equation.
y = log2 (x + 3)
The (x + 3) in parentheses in this equation causes the graph to shift three spaces to the left.
This graph is shifted left.
4. The fourth rule says that subtracting a number inside the logarithmic argument will cause the graph to shift right.
y = log2 (x - 3)
The (x - 3) in parentheses will cause the graph to shift three spaces to the right.
This graph is shifted right.
You can also have a graph that is translated in two directions. Take a look at this example:
y = log2 (x + 2) - 4
The graph of this equation will be shifted to the left two spaces and down four spaces.
This graph is shifted both left and down.
The Effect of Transformations on the Domain and Range
Earlier we discussed the domain and range of logarithmic functions and defined the domain as the possible x values and the range as the possible y values of a function. When the basic graph is transformed in a certain way, it will change the values for the domain and range of that function.
If the graph is shifted up or down, the domain will still be x > 0, and the range will stay y = all real numbers. If the graph shifts to the left or right, the range again will not change, but the domain will shift along with the graph. This is because we are moving the graph in the x direction, so the boundary line will change. If the equation is y = log2 (x + 2), then x can be less than zero - it just can't be less than or equal to -2.
If x = -2, then we get y = log2 (-2 + 2) which equals y = log2 (0) which is undefined.
All that to say that for the equation y = log2 (x + 2), the domain is x > -2.
The same rule applies if the graph is shifted to the right. If the equation is y = log (x - 3), then the domain is all numbers greater than 3, or x > 3, in order to keep the number being evaluated a positive number.
By adding or subtracting numbers from the logarithm equation or argument, you will shift the graph of the logarithm up, down, left or right. It's easy to do if you remember the rules of transformation. If the transformation is to the left or right, it will affect the domain of the graph but not the range. Up or down shifts will not affect the domain or the range of the graph.
After this lesson, you should be able to:
- Define logarithm and compare its graph to a graph of an exponential
- Explain how to graph a logarithm equation
- List the rules of transformation
- Identify how transformations affect the domain and range of the graph