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.
We often encounter exponential functions in science and mathematics. Exponential functions have a unique set of characteristics and asymptotic behavior that make their graphs recognizable. It is important to be able to recognize the graphs of exponential functions, the graphs of their inverses (logarithmic functions) and to know properties that help us solve equations with exponents and logs.
We're talking about exponential functions today and the first thing you need to know about an exponential function is the definition. An exponential function is one that can be written f of x equals b to the x. Where b is positive but not equal to 1, we call b the base. Now exponential functions come in all shapes and sizes and in order to see a bunch of them at once I'm going to show you a demonstration on geometer sketch pad. Alright so here we are on geometer sketch pad right now I've got function f of x equals 2 to the x showing that I can easily change that by pulling the slider changes up to 3 to the x, 4 to the x and one thing you'll notice is the higher b gets the steeper the graph gets. When b is lower the graph is less steep, b can also be less than 1 so for example when b equals 0.5 you get f of x equals 0.5 to the x. You have a decreasing exponential function we make b smaller still a third, a quarter and so on. The smaller we make b the steeper the graph gets, now you'll see why we don't let b equal to 1.
When b equals 1 we get a horizontal line, doesn't look exponential at all, so we want b to either be greater than 1 or be between 0 and 1 greater than 1 gives us increasing exponential functions between 0 and 1 gives us decreasing exponential functions. All exponential functions have some things in common one of them is that they pass through the point 0, 1 another one is when you plug in x equals 1, the y value is the base in this case 2 thirds and they all have as their domain all real numbers and as their range the positive numbers. You cannot get a negative output out of an exponential function and you can't get 0 out of it. So that's it in a nut shell, you got your increasing exponential functions, decreasing domain is all real numbers, range is the positive numbers.
Okay let's review what we just discovered, characteristics of exponential functions, first of all the domain is always the set of real numbers, the range is the set of positive numbers, they always have the y intercept 0, 1 and if you noticed they always have a horizontal asymptote y equals 0 asymptote. Let's graph 2 examples of exponential functions really quickly. y equals 2 to the x and y equals one third to the x, alright I want to graph them both on the same coordinate system. First I want to make a table, I'm going to plot a couple of points choosing easy values of x, right I want to make this easy to evaluate so negative 1, 0 and 1, 2 to the negative 1 is one half, 2 to the 0 is 1 and 2 to the 1 is 2.
Let me plot those right now, 2 to the 0 is 1, 2 to the negative 1 is one half, 2 to the 1 is 2, and then I can just draw a smooth curve connecting these points. Alright let's graph y equals one third to the x, first the table, we'll use the same values one third to the negative 1 is 3, one third to the 0 1, one third to the 1 is one third. So I'll plot those values negative 1 gives me 3 so up here 0 gives me 1, and 1 gives me a third. So let me draw my curve, okay and that's y equals one third to the x, this is y equals 2 to the x and so we have an example of an increasing exponential function and a decreasing exponential function.
It's good to remember an increasing exponential function the base will be bigger than 1, for a decreasing exponential function the base will be between 0 and 1.