Okay, hi everyone and welcome to our discussion on the Natural logarithms. This is a specific base of the logarithm that we are going to talk about, again there are infinitely many bases you can talk about, but this is a very specific one that will come up a lot in calculus. And so basically the natural log is when we talk about log to the base e, remember what e is. E is our number that's like 2 and change 2.718 blah blah blah blah blah is named after Euler. So natural log base e, this is a special base, it's so special it comes up so much that we abbreviate it and this is equal. It's the same as ln of x and this is what you see on calculators and everything like that. So natural log of x, so when you see ln remember it's a logarithm like every other log, but it is base e and therefore let's just get to know this thing. Let's just write down our stuff with this ln of x is why, so all the same rules, just different notation, it's good to see them. If and only if the base to y is equal to x or e to the y is equal to x, they are inverse functions, they cancel each other out. So they take, so ln of e the x, they can imagine them canceling as x or that's the same as e to the ln of x is equal to x. These are all things you're going to see a lot, we've seen them in the other general case. I just want to write them down with the specific natural logarithm case, when I have remember, we know that when you plug in the base to logarithm that you get one. What's the base with the natural log? It's e, so I see ln of e, that's the same base, so that's equal to one. You see that all the time, other things you'll see e to the zero, that's okay, remember e is just a number, it's like a number to the zero that's one. You'll see ln of 1 like any logarithm, if I plug in one I get zero and I'm just going to put this down 'cause you're going to see this a bunch, this is garbage. Ln of zero is undefined like any logarithm you can plug in zero, the graph of the natural log, you shouldn't have to think too hard about that. It is a base that is higher than one so it goes from low to high and it crosses right at one, so this is l equals ln of x, has domain just like every other ones. Has domains zero to infinity with a parentheses around zero, it has range all reels, has an asymptote as well at the Y axis. Just a couple things that I like to point out, its the natural logarithm, so why is it called ln? Shouldn't it be nl, shouldn't it be backwards natural logarithm, you ever think about this, anyone know the answer to this. And my hint is if you took Spanish or French or one of the other Italian romance based languages, if I have a red car, how do you say that? Usually you say, the car red, the adjective comes after the noun, so this is a little bit of French influence, they're doing the logarithm natural. So then your best French accent, logarithm natural, I took Spanish in high school, so can't help you there, but that's why it's backwards, so you're seeing a little bit as you go through this. You're going to see some German and French influences, a lot of that in your first couple years in math, a lot of the textbooks, a lot of the famous mathematicians were French or German. Euler was Swiss German, so you'll see a lot of that influence, but here's another one that you'll see it. I don't know if you ever thought about this, but something that I just like to point out, so the natural logarithm fight your English speaking American urges here. It is ln logarithm natural if you want to sound fancy an international, say backwards, you'll confuse everybody but you'll jus sound cool, ln of x. So just remember what this is, this is base e and one thing to just point out because if you have a calculator you'll probably. There's probably a button on there that does two things, at least on the standard ones, there's log of x. That is base 10, this 10 is a feel good number, we have 10 fingers 10 toes but it is no better and no worse than any other base. The best base for calculus that we're going to use is the natural log. So on your calculator if you doing that or on any calculator you see log with no base when it's blank, most often it means 10. There exceptions everything depending on your things, just be aware if I write ln of x then this is log base e of x. Log base 10, nobody likes that one, who cares? It is what it is. Log base e is the one that we like, okay? Remember or I don't know if you're free calc or whatever, but there is something called the change of base formula. It turns numbers from one base into another. I'm not going to stress it too much, I'm not going to talk about it too much, just know it exists. So that's one of the reasons why we talk about base e so much. There's so many options for bases and because you can convert one to the other, we might as well all just convert rs into a common calculus base, base e. So the natural log is the one that's going to come up the most. Be comfortable with it, be comfortable with this properties and be able to use it accordingly. So let's do an example where we actually solve an equation using this and you'll see what I mean. Okay, so here we go. Let's use a natural log to solve for x. Solve for x. Here we go, let's get fancy here. The natural log of the natural log of x is 1. Scary stuff two logarithm, I don't even like one logarithm, that's okay. So I need the x is trapped inside the logarithm. How do I do that? So it's called exponentiate? I need to get rid of this logarithm on front so what I'll do is I'll exponentiate both sides. You can think of that as raising e to the both sides. Either the ln x and the rules of algebra still apply, whatever you do to one side, you must do to the other side. So I raised, I took e to upstairs here I have to do either the one over here. You can't just do something one side and not the other side, that's illegal. So e ln they cancel their inverse functions, they go away and you just get ln x back. And e to the one some number to the one is just itself. So we have ln of x equals e and it's a little simpler. Went from two logarithms to one. It's nice, so let's do it again. Let's get rid of this ln in front of exponentiate both sides e to the ln x equals e to the e. You say e to the e is that allowed as that thing? Sure it's like 2 in change raised the 2 in change. If you plug it in your calculator you'll get a number back. If the calculator has no problem the number then neither should do. Once again, if you exponentiate both sides, the e ln cancel, that's their job. That's what they do. So you get e equals e raised to the e. It's kind of funny. It's kind of cute, it's kind of weird. I like it, okay? Let's do another one and see. If you want a decimal, if you're a decimal kind of person, go often. Plug into the calculator and do it. I am personally not a decimal person because I feel like if you write it to 4 decimal places you're wrong. Like e to e is the right answer. That is an infinite decimal expansion. That's the thing, if you round it to 4 decimals or 10 or 100, you're kind of getting approximation error. So I don't like that. But sometimes in the homework they'll ask you for like around this to the three decimals or whatever. So here we just want to sell for it. Let's do another one. 2 raised to the x- 5 = 3. Now, this may not look like logs are easy, but because the variables upstairs we're going to use properties of logarithms to solve it. In general, whenever the variables upstairs in the exponent you're going to use logarithms to solve this thing. Now the question is, which one do we use? I have basis here, I have base 2 and I have base 3. Which ones? This is where the question is answered immediately. Let's use the natural log. You could solve this to base 2, you could solve this to base 3. But because this is calculus course, let's just use natural log. So I'm going to take the natural log to both sides. Whatever you do to one side, you must do to the other. Can't just do with something to one side, not the other. Now I'm going to use properties of logarithms let's say exponents fall in front. Now, be careful here you must use parentheses. If you don't use parentheses, you're actually making a mistake. Moves that didn't work I'll try it again. (X- 5) time ln 2. Ln 2 is some number and if you want you know go punch it into the calculator. You got a decimal back, I would prefer not to work with decimals. So I'm going to leave it as ln 2, but you treat it like some number. You treat it just like some number, ln 3 is the same thing. So if I had like a 7 over here or pi or a 10 is whatever it is I want to solve for x, what do you do? You can divide by this some number. Okay, so that'll cancel and I get x- 5 = ln(3)/ln(2). Super important here is a place for a rookie mistake, folks will cancel the logarithms, but that doesn't make sense, because they're not numbers, they're functions. So ln of 3 over ln 2 is what it is. People try to combine this using a rule. If you check, there's no rule that deals with this. This is just number divided by number. Last but not least, you just add the 5 to the other side. So it's ln 3, whatever that is, some number divided by ln of 2 + 5. If you ever get down to a point when you can plug the answer to a calculator, I assume you're done. I assume you can take your fingers and punch some buttons. The algebra, the using of exponentials, the using of logarithms, that's the piece that I care about. That's the thing that I want to work with, okay? All right, let's do another one. So let's say, let's find the domain. Find the domain of f and f inverse. All right, so of course to do this, I have to give you an actual function. This is going to be one of our multi step questions, so here's the function, get ready. f of x is equal to the square root of 3- e to the 2x. So it's got a little e in there, no problem. We're going to use the inverse, it's probably going to use some logs most likely, so this is all you get. Find the domain of f and f inverse, doesn't matter which order we do first, but let's find the domain here of f, all right. What's going on with this one? 3 minus e to the two x. e has domain, all reals, 2x there's nothing wrong. There's no problems with the domain for e- 2x. There is something I have to worry about though is square roots and that is that I cannot plug in negative numbers. So I need my 3- e to the 2x to be greater than or equal to 0. Remember, it's okay to have 0 under a square root. The square root of 0 is just 0. So now I have an inequality with exponentials, everything with inequalities is still the same. Just gotta remember one little rule from back in the day. If you multiply by a negative, the sign, the direction of the inequality changes. Okay, so let's do some algebra with inequalities. So we'll add e to the 2x to both sides, adding, subtracting stuff, inequalities, that doesn't matter. Just one more thing to keep track of. So I have 3 greater than or equal to e to the 2x, that's fine. Now I take logarithms of both sides, so it turns out you can do this without a problem, so I'll take logs that does not change the sign. So ln of 3, remember that's some numbe,r natural log in e, they're inverse functions, they cancel, and I have ln of 3 is greater than equal to 2x. Lastly divide by 2, Now just be careful, we're dividing, but not by a negative, so I don't need to switch the sign. So last but not least, I have, we can write it this way, x is less than or equal to the natural log of 3 over 2. Just, and that you could be done here, but just the way sometimes if you check your answers in the back of the book or do something like this, just get used to seeing there's different ways. You can see it is 1/2 times the logarithm of 3, and then because it's a number in front, sometimes they bring it upstairs and it becomes ln of 3 to the 1/2. And you might even see this as ln of root 3. So however you want to write it x has to be less than that, so any number smaller than, less than or equal to this value is a perfectly fine thing for the domain. Okay, so that sort of did the part a, here comes the rest of our multi step. I need to find the domain of the inverse, but of course to do that I have to find the inverse. I have to find the inverse, so there asking me for a couple things. So let's do that over here. Okay, so remember four step process? Step one, write the function as y equals square root 3- e to the 2x. Step two, swap the xs and the ys, so it becomes 3- e to the 2y. Now comes the fun part, solve for y. y is trapped first and foremost under a square root. How do I free it? What do I do to undo a square root? I square both sides. Okay, let's move the e over, so I get e to the 2y equals 3- x squared. So just move some things around, that's okay. Remember, we're trying to solve for y here. It's a little counterintuitive, usually we're so used to solve for x, but I gotta solve for y. How do I do that? Well, it's trapped with an e in the base, so let's take ln to both sides. And then these guys cancel, so you get 2y is equal to ln 3- x squared. Do not distribute the natural log into both pieces, that breaks algebra. Log does not distribute. You can check your rules, hopefully have them handy. You're kind of stuck with this. It is what it is, what it is. So finally log is 1/2 times the natural To log 3 minus x squared. The last step in our four step process once we have solved for y is to write it as f inverse. Use the proper notation, so this is 1/2 ln 3 minus x squared. So there's the inverse function, and now of course we have to find the domain of this inverse function. This is not as bad, because remember the logarithm, the domain is positive numbers, I just need 3 minus x squared to be strictly greater than 0. Notice the difference, the last one under the root, you can equal 0, in the logarithm you cannot. Move the x squared the other side, you get 3 has to be greater than x squared. And there's a couple ways to see this x squared. There's our good old parabola. So the question is and if you put three on the map here, for what values of x is the parabola less than 3? So if you play around with this, you'll see it's at, remember this is the way you do algebra. You take plus or minus negative root 3 and positive root three. So x can be inside the interval, has the to be parentheses because it's not greater or less than equal to or greater than equal negative root 3 and then positive root 3. So, however you want to solve for that, gets you the answer there. So this is a multistep problem. It uses a bunch of information. It's a little trickier because it has inequalities. These are the kind of questions you're going to see, the tougher questions that you might see on tests or cumulative things like that. So go through this one, cover it up, start over again. If you didn't get this one right or confused about steps and then of course let me know if you have any questions. Okay, good job. See you next time.
