Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Today we are looking at exponential functions. In a moment we are going to do an example where we graph some exponential functions, but first I’d like to review some properties of exponents. You may remember this from your Algebra 2 class but it’s good to review them and you’ll use them a lot. So let’s go over them right now.
There are three of them I want to talk about. There is the product of power’s rule; b to the m times b to the n equals b to the m plus n. The quotient of power’s rule; b to the m divided by b to the n equals b to the m minus n. And the power to a power rule; b to then m to the n equals b to the m times n.
We’ll be using these all the times so it’s good to have them memorized. Here’s the example I want to talk about. Graph y equals 3 to the x and y equals 1/3 to the x how is this graph is related. I’m actually going to answer this question first because once I see how they are related it will be a lot easier to graph the two of them together.
I’m going to use my properties of exponents here. Let me start by looking at y equals 1/3 to the x. 1/3 is 3 to the -1. And by the third property of exponents, 3 to the -1 to the x, that’s 3 to the -x. So y equals 1/3 to the x, is the same as y equals 3 to the -x. What that means is, the graph of this function is a reflection of the graph of this function, around the y axis. Whenever you replace x in a function by -x, you get a reflection around the y axis.
And so, now that I know that, I can just graph this function and I know that the graph will be this will be reflection around the y axis. So let me start by graphing y equals 3 to the x, and I’ll make a table. I don’t need to plot a lot of points when you are graphing exponential functions because they have a nice predictable smooth shape.
I’m just going to plot three points. -1,0 and 1. 3 to the -1, 1/3, 3 to the 0, 1. 3 to the 1,is 3. So here are my three points. Let me plot those. I got (0,1) I’ve got (-1,1/3) and I’ve got (1,3). So I just connect these points and I've got my graph.
So that’s y equals 3 to the x. Now the graph of y equals 1/3 to the x is a reflection of this graph around the y axis. So all I need to do is take the points that I calculated and reflect those points around the y axis.
So the point (1,3) becomes (-1,3). The point (0,1) stays put. And the point (-1, 1/3) becomes (+1, 1/3). Then I just connect these three points. See how easy is to graph exponential functions if you use your properties. It’s really important to know that these two functions are reflections about the y axis of one another.
And what you do you, realize that you never need to be afraid of exponential functions with bases between 0 or 1. They are really just reflections of the other exponential functions.