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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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 **logarithmic function**, or the log function for short, is written as *f(x)* = *log* base*b* (*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* base*b* (*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.

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. 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.

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.

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.

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.

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.

If you picture the Richter scale, you can think of adding or subtracting from the log as increasing or decreasing the magnitude of the earthquake. Your graph will shift up if you add and down if you subtract.

If we added or subtracted from inside the log argument, our graph would shift sideways. If we add to the argument, our graph shifts that many spaces to the left. How does this happen? In our original function, *y* = *log* base2 (*x*), when *x* is 1, our *y* is 0 since 2^0 is 1.

Actually for all logs, when the argument is 1, then the function will equal 0. So, the argument actually gives us the *x*-intercept or where the graph crosses the *x*-axis. If the argument is just *x* itself, then the graph crosses the *x*-axis when *x* = 1.

If we added 4 to the argument so our function becomes *y* = *log* base2 (*x* + 4), then our graph will cross the *x*-axis when *x* + 4 = 1 or when *x* = -3. We subtracted 4 from sides to get *x* by itself. Comparing this to our original function, we see that this shifts the graph 4 points to the left. So, if we add to the argument, we will shift the graph to the left by that many spaces.

If we subtracted from the argument, then our graph would shift to the right that many spaces. We can calculate this point by figuring out when our argument will equal 1. So if we subtracted 4, we calculate *x* - 4 = 1 to find where our graph will cross the *x*-axis. We see that *x* = 5 when this happens. So we see our graph has shifted 4 spaces to the right.

Think of these types of shifts as shifting the location of the earthquake. Subtracting from the argument would take us further away from the location of the earthquake. The graph would shift to the right, taking the earthquake further away. If we add to the argument, we get closer to the earthquake and the graph shifts to the left, increasing the earthquake strength at every point.

What have we learned? We've learned that the **logarithmic function**, or the log function for short, is written as *f(x)* = *log* base*b* (*x*), where *b* is the base of the logarithm and *x* is greater than 0. We've learned that the basic graph of the log function has a curve in it. When we make changes to the graph, we see this curve flip, move up or down, or move sideways.

What are the rules? We see the graph flip over the *x*-axis when we add a negative sign in front of the log. The graph flips over the *y*-axis when we add a negative sign to the log argument. The graph moves down when we subtract from the function and up when we add to the function. It moves as many spaces as are added or subtracted. Likewise, if we add to the argument, we see the graph move to the left that many spaces. If we subtract from the argument, we see the graph move to the right that many spaces.

Watch this video lesson and increase your capacity to:

- Interpret a shift or flip in a graph with a logarithmic function
- Manipulate a logarithmic function to produce a flip or a vertical or horizontal shift in the graph

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- What Is an Exponential Function? 7:24
- Exponential Growth vs. Decay 8:41
- Transformation of Exponential Functions: Examples & Summary 5:51
- Using the Natural Base e: Definition & Overview 4:47
- What is a Logarithm? 5:23
- How to Evaluate Logarithms 6:45
- Writing the Inverse of Logarithmic Functions 7:09
- Exponentials, Logarithms & the Natural Log 8:36
- Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples 8:08
- Practice Problems for Logarithmic Properties 6:44
- How to Solve Logarithmic Equations 6:50
- Using the Change-of-Base Formula for Logarithms: Definition & Example 4:56
- How to Solve Exponential Equations 6:18
- Go to Exponential and Logarithmic Functions

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