It’s very interesting to see how different numbers affect the graph of the logarithmic function. Complete this lesson to see how changing the numbers in special places in the function either compresses the function or stretches it.

## A Logarithmic Graph

This lesson is all about changing the logarithmic function and its resultant graph. Mathematically, we define **logarithmic functions** as the inverse functions of exponential functions.

You will know when you are looking at a logarithmic function because you will see the log operator. The kinds of changes that we will be making to our logarithmic functions are horizontal and vertical stretching and compression. When we stretch a function, we make it bigger in a way. So, horizontal stretching means we make the function bigger horizontally.

When we compress a function, we make it smaller in a way. So, vertical compression means we make the function smaller vertically. Our general logarithmic function is of this form:

The *a* and *b* represent numbers. Remember, our log function can have different bases. We can have a log function to the second, third, or fourth base and so on. The standard base is base 10.

This is the log function that is used in calculators. For the log function where our *a* and *b* is 1, *f(x)* = log(*x*), we get a graph like this.

The red graph is the log(*x*) function, the blue graph the log(2*x*) function, the green graph the log(3*x*) function, and the purple graph the log(4*x*) function. Now, look at where these functions cross the *x*-axis.

Do you see that it is moving? As our *b* values become greater than 1, our *x*-intercepts decrease. It’s shrinking. The more we compress, the smaller our *x*-intercept is.

## Horizontal Stretching

If our *b* value is less than 1 but greater than 0, then we will have horizontal stretching. Let’s see what the graph does for log(*x*), log(*x*/2), log(*x*/3), and log(*x*/4).

Here we have log(*x*) as red, 2log(*x*) as blue, 3log(*x*) as green, and 4log(*x*) as purple.

What do you notice about these graphs that is different than the graphs for horizontal stretching and compressing? Do you see how the *x*-intercept remains the same for all the log functions? When we don’t change the *b* value of our log function, then our *x*-intercept won’t change. We can see that we have vertical stretching as our function goes higher and higher as we increase our *a* values to greater than 1.

## Vertical Compression

If you guessed that we get vertical compression when our *a* values are between 0 and 1, then you guessed right. Let’s graph the log functions log(*x*), (1/2)log(*x*), (1/3)log(*x*), and (1/4)log(*x*).

Our red is log(*x*), our blue is (1/2)log(*x*), our green is (1/3)log(*x*), and our purple is (1/4)log(*x*).

As our *a* values get smaller than 1 but still stay above 0, our log function gets smaller vertically.

## Combination

We can have log functions that use a combination of the above. The function (1/2)log(3*x*) for example has both a vertical compression and a horizontal compression.

We can summarize our stretching and compression information in this table. The *a* and *b* refer to the *a* and *b* values in our general logarithmic function.

Horizontal Compression | b ; 1 |

Horizontal Stretching | 0 ; b ; 1 |

Vertical Stretching | a ; 1 |

Vertical Compression | 0 ; a ; 1 |

We can combine this stretching and compression in four different ways in any log function.

- Vertical compression with horizontal compression
- Vertical compression with horizontal stretching
- Vertical stretching with horizontal compression
- Vertical stretching with horizontal stretching