The Number e: Study with Video Lessons, Practice Problems & Examples

Exponential functions with base e, approximately 2.71828, can be evaluated and graphed like any other exponential expressions. For example, to find \( f(2) = e^2 \) and \( f(-3) = e^{-3} \), use a calculator with the second ln function. The significance of e arises from compounding interest and appears in various applications, including population growth and radioactive decay. Understanding e enhances comprehension of exponential behavior in real-world scenarios, making it a crucial concept in mathematics.

The Number e

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 2 or one half or ten 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 we are going to work with this function with 2 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, and 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(x)=ex, and go ahead and evaluate this for x equals 2. Now I'm simply going to plug 2 in for x into my function, so f(2)=e2. Now when 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 2, I would type second ln and then 2 in order to get my answer, rounding to the nearest hundredths 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 3. Now, again, we're just going to be plugging in negative 3 for x here. So f(-3)=e-3, which knowing our rules for exponents, this would really just be 1/e3. 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 3, I would type second ln and then negative 3. And rounding to the nearest hundredths 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(x)=ex. And you see here that it's right in between my graphs of 2 to the power of x and 3 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 23. So I know that my number e is right in between 23, so it makes sense that the graph of my function e to the power of x is right in between the graphs of the x of the exponential functions with base 2 and base 3. 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 used for any other more complicated functions of different bases like 2 or 3. 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 ert right here. And we want 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-lives. 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.

Graph the given function.

$g\left(x\right)=e^{x+3}-1$

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