Hey, everyone. We just learned about a frequently occurring log, log base 10. And there's actually another log that occurs rather frequently, log base e, called the natural log. Now don't worry. We're not going to have to learn how to do anything new here. Here, I'm going to walk you through how we can treat log base e just as we would any other log of any other base, just with special notation. So let's go ahead and get into it. Now, log base 10, our common log, we would write as log, and similarly, log base e gets its own special notation. It's always written as ln. Now, a way to remember that is because this is the natural log, if we take the first letter of each of those words, natural log, and we reverse them, we get ln, which we're always going to read out as natural log. Now the natural log also has its own special button on your calculator, the ln button, for whenever you need to use that to evaluate something on your calculator.
We just saw that whenever we have an exponential equation b^x = m, we can rewrite this in its log form as log_b(m) = x. Now the same thing is true of anything with a base e. We treat it just like any other base. So if I have e^x = m, I can rewrite that in its log form as log_e(m) = x. Now one thing we just need to be careful of here is that this log base e should always be rewritten as simply ln. You're pretty much never going to see log base e written out like that. It's always going to be written as ln. So with that in mind, let's walk through these examples together. Now, our first example here is x = ln(17). So here, let's go ahead and actually rewrite this as log base e to help this make a little more sense to us. So here we have x = log_e(17). And we want to rewrite this in its exponential form because it's currently a log. So we're going to start at the same place we always do with that base. So here, our base is e. So starting with that base and then circling to the other side of my equal sign, e^x, and then circling back to my original side of my equal sign is equal to 17. And that's my final answer here, e^x = 17, just like we would with any other base.
So let's look at one final example here. We have e^x = 4, and we want to rewrite this in its log form. So, again, we're going to start at the same place with our base. So here, this base of e becomes the base of my log. So log_e(4) circle to the other side of your equal sign, and then circle back is equal to x. Now we're not quite done here because, remember, we need to make sure we have the correct notation going on in our final answer. So we know that this log base e is really just ln, the natural log. So this is really ln(4) = x, and here's my final answer here.
So now that we know a bit more about the natural log, let's get a bit more practice. Thanks for watching, and let me know if you have questions.