Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples

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  • 0:01 The Basic Graph
  • 2:22 Flips
  • 4:10 Vertical Shifts
  • 5:01 Horizontal Shifts
  • 7:03 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson and you will see what the basic graph of the logarithmic function looks like. You will also be able to identify the kind of shifts that will occur by just looking at a logarithmic function when you finish the video.

The Basic Graph

The logarithmic function, or the log function for short, is written as f(x) = log baseb (x), where b is the base of the logarithm and x is greater than 0. What does this mean? It is telling you what exponent, f(x), is required to raise the base, b, to the number x. We can rewrite this relationship using the exponential form so that y = log baseb (x) becomes x = b^y. You can see that the base of the log, b, gives you the base of the exponent.

When graphing log functions, we can use this information to help us manually calculate points on the graph. For example, say we wanted to graph the function y = log base2 (x), where the base is 2. We can calculate points using the exponential form. We do this by plugging in different values for the exponent, y and using the exponential form to find x.

We begin with the number 1 for y. We find that 2^1 = 2. So that means the x is 2 and the y is 1. Rewriting this in the log form we have, 1 = log base2 (2). From this we already have one point, (2, 1). For another point, we can calculate 2^2, which equals 4. Our x is 4 and our y is 2. In the log form, it is 2 = log base2 (4). Our second point is (4, 2). We can continue in this manner to get even more points. Graphing this function, we see that this log function has a general curve.

The basic graph.
log graph

All log functions will have a curve that looks like this one. An example of a real-world log function is the Richter scale used to measure the magnitude of earthquakes. For each number increase, the earthquake is actually 10 times stronger. The Richter scale uses 10 as the base. When the function is changed in various ways though, this graph will also be shifted in various ways. But the shift or change that happens is predictable. Let's go over the three different kinds of changes that we will see.


The first is when we have a negative sign. When this happens, our graph will flip, either over the y-axis or over the x-axis. The axis that the graph flips over depends on where the negative sign is. When the negative sign is inside the argument for the log function, the graph flips over the y-axis. (We know we cannot take the logarithm of a negative number, but by flipping the graph over the y-axis it changes all the x values to -x. When we put a negative value into log (-x) we get log (- -x), a positive value, so we are actually taking the logarithm of a positive value.) So, for the function y = log base2 (-x), we will see that our graph has changed to the mirror image when the mirror is the y-axis. We see the point (4, 2) becoming point (-4, 2), the point (2, 1) becoming point (-2, 1), and so on.

Flip across the y-axis.
log graph

When the negative sign is in front of the log, then we will see that the graph becomes the mirror image when the x-axis is the mirror. So, for the function y = -log base2 (x), we see the graph flips over the x-axis. We see the point (4, 2) becoming the point (4, -2), the point (2, 1) becoming the point (2, -1), and so on.

Flip across the x-axis.
log graph

One way to remember the flips is to ask yourself which letter the negative sign is closest to. If it is the x, then the flip is over the y-axis since we are switching sides across the y-axis. Positive x values become negative and negative x values become positive, thus switching sides. If the negative sign is not next to the x inside the argument, the flip is over the x-axis since this negative changes the output of the log, which becomes the number on the y-axis. So positive y values become negative and vice versa.

Vertical Shifts

We will see up and down shifts when we add or subtract from our function.

When we add a number to our function, we will see our graph go up by that many spaces. For example, say we add a 4 to our function so our function becomes y = log base2 (x) + 4. We will see our graph shift up by 4 spaces. Think of it as adding 4 spaces to every single point of our original graph.

Shift up.
log graph

If we subtracted 4 from our function, we will see the opposite. We will see our graph shifting downwards by 4 spaces. Think of it as subtracting 4 from every single point of our original graph.

Shift down.
log graph

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