The Number e - Video Tutorials & Practice Problems
On a tight schedule?
Get a 10 bullets summary of the topic
1
concept
The Number e
Video duration:
4m
Play a video:
Hey, everyone, when working through problems with different exponential expressions and exponential functions, you may come across one that doesn't have a base of something like two or one half or 10 or any other number, but actually has a base of E this lowercase E. And the first time you see that you might be wondering why there is another letter in that function when we already have X right there. And how are we going to work with this function with two different variables? But you don't have to worry about any of that because here, I'm going to show you how E is literally just a number. We can treat it just like we would any other exponential function and evaluate and graph it using all of the tools that we already know. So let's go ahead and get started. Now, like I said, E is not a variable at all but simply a number. And another similar number that you might be a bit more familiar with is pi we know that pi is 3.1415 and so on this long decimal that we don't write out, we just write pi E is really similar. It's this long decimal 2.71828 and so on. But we simply write it as E now because it's a number, we can treat it just like we would any other exponential function and do things like evaluate it for different values of X. So let's take a look at our function here. F of X is equal to E to the power of X and go ahead and evaluate this for X equals two. Now, I'm simply going to plug two in for X into my function. So F of two is going to give me E to the power of two. Now, in working with exponential functions with base E, we do want to use a calculator to evaluate these and the buttons that you're going to use on your calculator in order to get this base E are second Ln, this should give you E raised to the power of and then you simply type in what your power is. So for E to the power of two, I would type second Ln and then two in order to get my answer rounding to the nearest hundreds place which would be 7.39. Now this would be my final answer here. But let's go ahead and evaluate this function for X equals negative three. Now again, we're just going to be plugging in negative three for X here. So F of negative three is simply E to the power of negative three, which knowing our rules for exponents, this would really just be one over e to the power of positive three. And you can type either of these into your calculator and you should get the same answer. So typing E to the power of negative three, I would type second Ln and then negative three and rounding to the nearest hundreds place I would get an answer of 0.05 as my final answer here. Now we can evaluate exponential functions of base E like any other exponential function. And we can also graph them just like we would any other exponential function as well. So if I take my function here E to the power of X, I know that my graph is going to have the exact same shape as any other exponential function. So my graph is going to end up looking something like this F of X is equal to E to the power of X. And you see here that it's right in between my graphs of two to the power of X and three to the power of X which this happens because E we know is this number 2.718 and so on, which happens to be right in between two and three. So I know that my number E is right in between two and three. So it makes sense that the graph of my function E to the power of X is right in between the graphs of the, of the exponential functions with base two and base three. Now, if you're faced with graphing a more complicated function of base E, you can simply graph it using transformations the same method that we use for any other more complicated functions of different bases like two or three. Now, hopefully with all of this, you see that we can treat exponential functions of base E just like any other exponential function no matter what the scenario is. But you still might be wondering why we need this base of E in the first place. Why do we need to have this base? E when we have all of these other numbers to choose from? That aren't crazy decimals. So I'm going to give you a little bit more information about what E is and where it comes from. So E actually comes from the idea of compounding interest, which is this equation right here and we want our co our interest to compound as much as possible. So if I take the number of times that my interest is compounded this N and take this all the way up to infinity. This equation is going to end up giving me 2.71828 and so on which we know is just E, now, that's where E comes from compounding interest, but it's actually going to pop up in a ton of other stuff that you'll see. Now you might see in your other courses that E is a part of predicting population growth and working with something like radioactive decay and half life. So E is just a number, but it's a number that describes a bunch of different things going on in the world. And even with describing all of these different things and being super useful, we can treat it just like we would any other exponential function. So with that in mind, hopefully you have a better idea of what E is and why we need it and how exactly to work with it. Thanks for watching and let me know if you have any questions.
