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Interactive video lesson plan for: Master Graphing Logarithmic equations with reflections then determine the domain and range

Activity overview:

Subscribe! Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out. Welcome, ladies and gentlemen. So what I'd like to do is show you how to graph exponential equations, I'm sorry, when they're in the form of y equals a times log base b to x, as well as y equals a times ln of x. And basically, these are going to have some very small trans-- well, there's actually not going to be any-- the only transformations we're going to have is reflections.

But I kind of want to talk a little bit as far as the bases of the logarithm graph as well as what the general parent graph looks like because a lot of students get confused, especially once we start throwing in numbers in different places. Even though they're not translations, we're not shifting the graph left to right, up or down, they are going to have some effects. Now I don't have any graph paper. I am just doing a general idea of what the graph is just using the basic transformations.

So if you want to kind of see how a graph changes, for instance, if it's log base 2 or if it's log base 4, that is going to affect the graph but I'm just not going to get into it. So I do I just recommend getting into the graphing technology you can see, but what I'm going to be dealing with is the points on the graph that are going to be ever changing. It's not going to matter what the base is.

The points in the transformations that I'm going to deal with is still going to produce a graph that's going to be very similar, even if those numbers are different. For instance, same thing even if I had a number 3 here. If it was a 3 or a 13, the graph's going to look different, but I'm just going to try to-- but the points that I'm going to provide are going to be what the graph looks like so it will give you general good idea.

All right. But before I get into that, I would like to kind of go over the general parent graph for you so you can-- so we can get kind of an idea of what the logarithmic graph looks like. And the best way to do that is to kind of go back to our exponential equation. Let's just do y equals 3 to the x.

If you remember when we were graphing the exponential equation, we always crossed at our a, which in this case was 1. And the graph looked something like that. Dot, dot, dot, dot, dot. Had of nice little asymptote, right? Well, in my previous video, I talked about how exponential logarithmic equations are inverses of each other. That means they can be reflected about the y equals x line.

Now, for the graphs that we're going to be working with today, our a is technically always going to be-- not our a is always going to be 1, but the y-intercept is always going to be at 1. So we're not going to be dealing with any functions where it's going to be different until-- unless we get to transformations. So the graph of a logarithmic equation it's going to look something like this. OK? So it's basically a direct reflection across what we call the y equals x line, which is this line right here.

All right? So there's a couple important characteristics. We have a x-intercept at x equals 1. So that coordinate point is 1 comma 0. And we also now have an asymptote. It's a vertical asymptote. So where we originally had a horizontal asymptote by reflecting over the y equals x line, we now have a vertical asymptote. And that's very important, especially for identifying the domain and the range for the graph.

So that's kind of like the general idea what the graph looks like. What I did in exponential, though, is I created a table, which I'd like to kind of do as well. And the main thing is for all of our graphs until we get to transformations, this is going to be our parent graph. And the only thing really that changes with-- if you have a number on the outside, or if you have a different number in the base, that's just going to change how the graph increases.

It can be really sharp, it can be much more shallow, and so forth, but I'm not going to get into details with that because I don't have a graphing calculator-- or I mean, I don't have graph paper. And really, the main important thing is I want to produce a graph that's going to be close and similar that you can be able to identify it and understand what exactly we did to obtain that.

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