Transcript for:
Logarithms and Exponential Functions

hey welcome to another video we gotta get on today we gotta finally put things together and go from exponentials to something called logarithms not just how to do it i'm gonna show you why i'm gonna show you why this stuff comes about where logarithms come from and how they are inexorably linked together where you can't get away from these things that are joined because they're inverses we're going to learn about that today right now so what i intend to do in this video is teach you where a logarithm comes from why they have to exist what the graph looks like then because we know the graph looks like it because they exist we'll see what the structure is from an exponential trying to solve for an inverse after that we'll talk about the domain how to find the domain of a logarithm how the argument always has to be positive why that is graphically and then we'll graph just a few examples to get the feel for how transformations would work with logarithms so here we go let's take a look back at what exponentials are something that we should be very familiar with we've seen this in a couple of classes now and we see this in a couple videos ago that these exponential functions whether a is above one or below one creates something has some key points that we understand some domain that we understand some range that we understand so let's take a look at that whenever we get an exponential function we understand that before we shift around before anything like that we have some some very important features we understand that every one of these exponentials goes to the point 0 every single one of those is going to go through that point because when we evaluate for 0 any base constant base to the 0 power is going to give us 1. we also know that they all go through the point 1 comma a we know this because if we evaluate for one then the base to the first power gives us back that base so one comma a if a is above one if a is more than one then one comma a will be vertically more than this point and if a is less than one vertically one comma a will be less than this point which is what gives us this increasing or decreasing graph for exponentials is why they look a little bit different we also know that each and every one of these has a horizontal asymptote somewhere if we haven't shifted it it's right on the x-axis y equals zero that's important we know that the domain we can plug in anything anything to an exponential and we have no domain issues whatsoever so from negative infinity to positive infinity this thing is continuous and has no domain restrictions whatsoever however if you take a base that's positive and you take it to any exponent positive or negative you can't ever get out a negative so our range is not all real numbers our range is from zero to infinity but this is important this is something to focus on because when we get logarithms the domain of logarithms we're going to see that you cannot have zero in the argument y stems from right here you cannot ever get out a zero in an exponential equation or sorry function this is because this never actually touches zero you can't ever touch you can't cross it we can't ever get out negatives or zero from an exponential what that's going to turn into is we can't ever plug in negatives or have an argument that is negative or zero in a logarithm and see why that is so that's a recap of what exponentials are now here's the main point main point of an exponential is if you take a look at that graph and apply the horizontal line test this thing is not only a function but a one-to-one function and we learned something in inverses that every one-to-one function has to have an inverse what's that mean that means that this function which is one to one and this function which is a one to one they're both exponentials also have to have inverses so we thought about that we said okay well what's the graph of an of an inverse really look like if this is an exponential it's one to one it has an inverse we should be able to graph it if the graph of this exponential looks like this then the inverse is going to be a reflection across the y equals x line well that's true and in fact we know how to find inverses just by reversing the x and y coordinates you need that part of that that lesson when we talked about inverses to to get this right now so main concepts are every one to one function has an inverse the graph of an inverse is the reflection across the y equals x line how to find points is to reverse the x with the y so get this we know exponentials have an inverse pretend we didn't even name this as a logarithm yet someone somewhere would have said okay that's a one-to-one function great it's got to have inverse fantastic we can graph this inverse all right what are we going to call it i don't know um and we called it a logarithm a long time ago when these things were discovered and so when we take a look at this as a one-to-one function that has an inverse if we want to reverse our key points it's going to give us a really solid graph of a logarithm check this out if every exponential has 0 1 then every inverse of an exponential we're going to call this a logarithm in a minute every inverse of an exponential is going to have the point 1 0. if every exponential has this point 1 comma a then every inverse of an exponential is going to have this point a comma 1. now here's where it's important understand that our a is more than one if this is going to have the point a comma one if a is more than one then a is going to be located over here somewhere this is the point a comma one where our a is more than one what if they had been less than one it would be over here and we're going to see that on that graph over there and this is the point one comma zero one last thing if your exponential has a horizontal asymptote and we're switching x and y so think about this if you have a horizontal asymptote at y equals zero then if you switch x and y you'll have a vertical asymptote at x equals zero and that's also true more than that if this is a horizontal asymptote on let's call it the negative x-axis then this would have a vertical asymptote on let's call it the negative y-axis and we can graph this if we think through that this is a reflection across y y equals x then this exponential which is one-to-one which has to have an inverse which is found by switching x and y and along with our horizontal into a vertical asymptote this graph has to look like this and that's about as close as i can make it to represent the inverse of an exponential so here is the big idea this was an exponential that had to have an inverse and we just crafted so we knew it had to because it was a one-to-one function had to have an inverse function we just did it we said all right yeah reflection across y equals x let's switch some key points 0 1 becomes 1 0 for the inverse 1 comma a becomes a comma 1 for the inverse horizontal asymptote on the negative x axis becomes vertical asymptote on the negative y axis cool if a is more than one then a should have a key point more than one comma one this is the graph of the inverse of an exponential we know it exists we even have a picture of it we can do the same thing over here i'm gonna move a little bit quicker but but check it out if this is a one-to-one function it's also an exponential just happens to have a less than one every exponential has key points of zero one and one comma a why this decreases because one comma a where a is less than one has an output below one that's why that looks the way that it does so if that's one-to-one it also has to have an inverse and the graph of that inverse has to be reflection across y equals x we also have to be able to reverse our points so x becomes y y becomes x and the horizontal asymptote changes to a vertical all right well what would that look like if this graph has 0 1 then the inverse should have 1 0. please notice as we're going through this we're getting exactly the same points it's going to look different because the a is less than 1 but both of these have a key point at 0 1 for exponentials and therefore one zero for the inverse so one zero if this has a key point of one comma a then we're gonna have a key point and inverse of a comma one but because a is less than one if the key point is a comma 1 and a is less than 1 it's going to be between 0 and 1. this would be the point a comma 1 and this is the point one comma zero so zero one becomes one zero for the inverse one a becomes a one for the inverse except a is less than one so it's gotta be it's gotta be positive right it's gotta be somewhere between zero and one and give us out one also one more thing this is going to look a little funny because this original function our exponential crosses the y equals x line well what that means is that the inverse is also going to cross the y equals x line and it's going to be at the point where they intersect because that point of intersection when you switch x with y would give you the same input and output it would cause that to just reflect and still be on that that that line of symmetry finally if we have a horizontal uh asymptote at y equals zero so on the positive x-axis then we're going to have a vertical asymptote for the inverse on the positive axis and here's what this graph looks like they look similar because you have a graph that is being reflected but like almost has the same sort of shape but notice how they are distinctly different your exponential decreases and your logarithm does decrease but they have different things about them your exponential you still have a domain of all real numbers you can plug in anything in this function no problem and you have a range from zero to infinity no problem you only get out positives but look at your logarithm your logarithm has a domain that's purely positive you cannot have an argument that is negative or even zero but you can get out all real numbers your domain and your range i hope that this makes sense to you are also going to switch everything about a function and its inverse switch domain becomes range range becomes domain y's become x's x's become wise horizontal asymptotes become vertical vertical become horizontal that's what happens with inverses your key points reverse themselves and we're going to see that right now so this is the graph of our our logarithm as it comes from this one-to-one function called an exponential so we we took a look at it said exponentials are one to one they have to have inverses and we graph those inverses if that's the case if we if we know that our graphs look like this and every single one of these logarithms is going to have a vertical asymptote and every single one of these these inverses of an exponential let's call it have a a key point at zero one i sorry say that one that wrong at one zero because it came from key point zero one so at one zero and if every one of these inverses of an exponential has a key point of a comma one in this case a is more than one and in this case a is less than one then every one of our logarithms is going to look inverses of an exponential is going to look one of these two ways for a that is greater than one with this vertical asymptote if a is more than one we get this picture it's always increasing if a is less than one we get this picture always decreasing very very much like our exponentials exponentials were always increasing or always decreasing depending on whether a was greater than 1 or less than 1. i hope that i've explained this in a way that makes sense to you i hope that i've shown you the key points many people do not really fully grasp where a logarithm comes from where the inverse of an exponential comes from and is called a logarithm so they just kind of go through math and go oh there's things called exponentials there's this thing called logarithms and they miss the point they just pass it right by they don't get it what i try to do in the beginning of this video is show you where they come from so we're about to call the inverse of this exponential a logarithm so we'll do that a minute but for right now i wanted you to understand that the main points the main points were every exponential is one to one and therefore had to have an inverse we found the inverse we took the key points reversed them domain became range horizontal became vertical asymptotes with a greater than one we had an increasing and then still increasing graph but it was reflected across y equals x and the decreasing was still decreasing but reflected across y equals x we know a lot about this we know that for every single one of our inverses we'll be able to say what the domain is what the range is what the key points are but we have to call it something if i were to do this and say that's a one-to-one function and we should be able to find the inverse how do we find inverses algebraically well we we replace it with y we switch our y into an x this is better this god makes sense to you like we've done this right we know this this is how we find inverses and x becomes y so we'd say the thought process long time ago was one to one one to one here's the graph of it no problem here's the graph of this one no problem if we're supposed to be able to find an inverse so we should be able to do it algebraically so function replace that with y the y changes to an x the x changes to a y and then what are we supposed to do well we're supposed to solve for y and you'd get here and you'd go yeah great um how do i how i'm supposed to solve for y i can't divide i can't multiply i can't add i can't subtract uh i can't would you take a y th root or something on both sides but then your variable's over here somewhere like i don't i don't want to put a radical with the index of y it might be really really stinky awkward like we don't want to do that so how in the world am i supposed to solve for y which is what is necessary to find the algebraic representation of the inverse function here we have to solve for y and so what people did is they said i know this thing exists we just proved it graphically we know the horizontal line test says we have to have an inverse but there's nothing mathematically that we have that would solve for y already a logarithm says if you have an exponential you can change this into what is called logarithmic notation and so we would say here's what logarithms do they solve for your exponent in some sort of an exponential that's all they do what a logarithm does is it separates the base from the exponent and exponential notation puts it back together so if we have an exponential what your logarithm is going to do it's going to say here's your exponent your logarithm is going to separate your base from your exponent notice where the exponent went where the base went basement is what's called a base for a logarithm it's the same word it's the same base this is what a logarithm does this is the inverse of some exponential with the same base some really fascinating things happen with logarithms such as when you compose it with an exponential they cancel if they have the same base we're going to see all these properties as we move forward into logarithms but this is the main concept of what they what they do so i'm going to write out now some of the key things for for logarithms we'll see how this stuff switches around and then we'll get into domains and graphing okay so here we go when we whenever we find an inverse from a root function what happens is all of our x and our y junk they they switch and so if we take a look at our graphs and we acknowledge what our exponentials have we're going to see a lot of similarities but they're they're reversed what i mean by this is your domain for an exponential was all real numbers nothing you couldn't plug in that's going to be your range for your logarithms notice how we can get out our ranges from negative infinity to positive infinity this will eventually slowly get to positive infinity same thing as as this our range here is from negative infinity to positive infinity this will eventually cover the entirety of the y-axis so our domain for our original function becomes our range for our inverse our domain for exponentials become our range for our logarithms so we get out all real numbers for our range now take a look at your exponential our exponential said you can't get out everything for an exponential in fact the outputs only cover from zero to infinity but not including that zero due to the fact we had a horizontal asymptote okay what that what's that mean for the inverse for the logarithm if we can't get out everything we can't plug in everything for the logarithm of the inverse we don't get out anything but 0 to infinity that means that the argument inside of our logarithm can't ever be anything but 0 to infinity we cannot have negatives you cannot even have zero our domain here is zero to infinity take a look at what we can plug in there is nothing on this side of the x-axis you cannot plug in a negative oh look at that vertical asymptote you can't even plug in zero we know all about vertical asymptotes now you can't even allow this this is unallowable it's undefined you cannot have zero as an input of your logarithm it is undefined in both of these cases they both have vertical asymptotes and they both only have inputs that are allowed to be on the positive x-axis that's okay that comes from the domain of our exponentials it's expected now our key points we saw this as we're going through the graphene key points here are 0 1 2.4 both of your logarithms are 1 0. key points for your exponential are 1 comma a switch that around key points for your logins are a comma 1. the reason why these graphs look different is because of this point they both go through one zero just like both of your exponentials went through zero one they both go through the point a comma one a comma one just like your exponentials went through one comma a but the difference becomes if your a is more than one then a comma one is a point greater than one up to one and if a is less than one this is a point a is less than one up to one so less than one up to one that's why they increase and why they decrease lastly because our original functions or exponentials had a horizontal asymptote at y equals zero these are going to have a vertical asymptote at x equals zero and that's about it the main the key point that i need you to know right now is that these graphs come from exponentials logarithms are the inverse of some exponential with the same base a huge point the next thing i need you to understand is the domain you may never plug in a value that causes the argument we said we call that the the inside of your logarithm sometimes the argument we can never allow an input that causes that to be negative or even zero it must be strictly positive it's a very optimistic function it's always positive you can only plug in values that cause this thing to be positive okay that that's a must it must do now the outputs can be negative sure absolutely the outputs can but the inputs must always cause the argument what's inside the logarithm to be positive we're going to practice finding domain but those are some notes that you're going to want to make before we can talk about graphs before we can talk even about domain we do need to understand how logarithms work how we can change from exponentials to logarithms and logarithms exponentials this will get you used to the idea of what a logarithm does and what a logarithm looks like so what we learned in just the first part of this video is that exponentials do have an inverse and we call that a logarithm what should they do they should undo each other so here's what i want in your head this is very important this is what i want you to understand what a logarithm will do is it will take an exponential and it will pull it apart so it will take the base and the exponent and put them on opposite sides of an equation it will solve for the exponent that's what logarithm does it separates the base from the exponent what an exponential will do is take a logarithm and put it back together it will take the base and exponent and put it in an exponential notation so we can go backwards and we can go forwards on these we're going to practice that right now we'll also learn about something called a natural log and a common log if you haven't seen them or if you've just forgotten about them so here's what happens if we have some exponential notation i'm going to walk through this very quickly this should be review for you but i'm just doing the courtesy of getting used to it again so that way i can talk about it at length so this and this and this and this and this would all be called exponential notation so when we look at it you have a base to an exponent equals some sort of an argument here so 3 squared is 9. or a to the third is 2.1 2 to the x is 7.2 our bases are 3 a and 2. 10 and e these are still exponential notation a base which is a constant raised to some sort of a power equals some number and same thing here 10 is our base raised to an um a power equals 2.3 what a logarithm will do is write this in a different notation and it will separate the base from the exponent so when we write these as logarithms here's what happens base gets written with the logarithm so log base 3 is how we say that not log of 3 that's a different um a different connotation and that's going to stand for something like a common log which we'll get in a second so we'd say log base 3 it's written as a subscript it's not on the same level as a log it's a subscript it's slightly below if you were to draw a straight line it looks like that log base three of nine equals two the nine and the two are the same level as the law of log base think about like in the basement log base three of nine equals two what has happened is the logarithm has separated the base from the exponent and put it on opposite sides of your equation and the 9 becomes the argument of the logarithm it says something a little bit funny it says it says what an exponential is sort backwards so what this exponential says is 3 to the second power is 9. what the logarithm says is 3 raised to what power will give you 9. or in order to get from 3 to 9 what power do you need well you need to why is it is strangely worded well that's kind of that that's how inverses work in general so you can say uh three plus five equals eight you can say that and that's probably the most straightforward way to say it but you could also say it with an inverse you could say 3 minus negative 5 is also 8. well that is true and that's using some inverses internet so that this is the inverse of addition and this is the opposite of five well you're you're so you're doing something along the same lines here logarithms will say the same thing but they'll say it in a different manner so because their inverse is almost like an opposite idea here 3 squared is 9 or 3 to 9 needs a power 2. or in order to get from 3 to 9 what power do i need i need power 2. so let's move on a little bit let's try to write all of these in logarithmic notation if i have a to the third power equals 2.1 our base is a our exponent is 3. what a logarithm is going to do is rip those things apart so let's say log base a that is our base for the exponential that is our base for our logarithmic notation our exponents three logarithms always separate those things and two point one becomes argument so log base a of two point one equals three start saying in your head as you're writing these things down to get used to the way we say these we don't want to say log of a of 2.1 because that log of is always going to have a base of 10. and we're going to see that when we get to common logs so we're going to say log base for other bases of whatever argument is equals whatever the exponent is logarithm separate based on exponent you should try this one and this one on your own right now if you haven't do that or do that right now so this it says 2 to the x equals 7.2 we know what our bases are base is 2. our exponent is x this is the standard type of exponential right these are a little bit not true exponentials there's exponential notation we're kind of moving around but this is a true exponential it has a base that's a constant a variable that is an exponent watch what happens when we put this into logarithmic logarithmic notation this is going to solve for our variable for us that's what it's supposed to do an inverse should undo its original root function so a logarithm should undo the exponential that's why a logarithm rips apart the base from the exponent it's going to put them on opposite sides of the equation it's going to solve for your exponent so with our log having a base of 2 from our exponential having a base of 2 our exponent of x your logarithm is going to put your base and your exponent on opposite sides and then it's going to have an argument of whatever your exponent whatever your exponential is equal to so log base 2 of 7.2 equals x let's solve for x for us that's what a logarithm does it will always separate your base from your exponent and put your exponent by itself on one side of your equation it'll always take what your exponential is equal to and treat it as an argument inside your logarithm your calculators will do this in a manner of speaking most of them have a base of 10 some of them like some casios i know of will allow you to plug in both a base and an argument some don't let you do that some will just have a base of 10 or base of b and i will give you a change of base formula to show you how that works but that's it now this other one well this is also an exponential like we we are used to where it has a base of 10 and x root of x if we put this in logarithmic notation where we let the logarithm separate the base from the exponent our base is 10 it's written as a subscript subscript log and 2.3 and x are on the same level you should be able to draw a line through that have them all be on this to that same line this is solved for x well that's great that's what an inverse should do it should solve stuff for us a logarithm will solve an exponential an exponential will solve a logarithm we're going to get to that we start solving these things it's going to be awesome but for right now we're just we're practicing and we're seeing this we're just getting used to it your calculator will do this the log button on your calculator will allow you to plug in 2.3 you can try it right now if you want to the log button on your calculator has a base of 10. in fact that base of 10 because we have a base 10 number system is so common we call it a common log this is a common log and in fact whenever you have a base of 10 you don't even write it when you say this you don't have to say log log base 10 you just say log of 2.3 and people will understand that you're talking about a base of 10 because this common log is log of 2.3 means we have a common log we have a base of 10. that's why it's so important to say log base whatever this is log base 2 log base a log base whatever log base 3 but here when we say log of 2.3 we are implying that that has a base of 10. this is called a common log so whenever you see log and missing your base it's not missing it it's just that you need to know that's a base of 10. the same thing happens here this is also an exponential it has a base that is a constant it's basis e that's 2.7 ish and then it's a number that doesn't end doesn't repeat for any length of time and we can use a logarithm to write this in logarithmic notation logarithms always do the same thing they take your exponential and they separate the base from the exponent for any true exponential it will solve for your exponent and let you solve that here it's already solved we know x equals this thing now what is this thing well it's not a variable it stands for a number it is a constant here log base e of 8 equals x says this is some sort of number and your calculator will actually handle it but you're not going to see log base e on your calculator written this way in fact you'll never see it again because this is called what is called this is what is called a natural log why is it called natural log because it's a log with the base of the natural number e called the euler number or the natural number is a constant that we call the natural number it's it's just e we just write it as e but if you ever see that base in a logarithm or have a logarithm with that base you're going to write it as the log with a base of a natural number or a natural log how we write that is with this ln so if you ever see or say or hear ln8 ln of 8 natural log of 8 what it says is you have a logarithm with a base of e please understand this is still a logarithm it is just like this okay it's a logarithm with the base of e we just don't write it if you ever have a base of it you're going to write ln that stands for natural log if you ever write any sort of if you ever see any ln and you change the exponential notation this has a base of e just like that has a base of 10. they're not missing they're just written certain ways because they happen so often that we get tired of writing e it happens actually happens probably more often than that one natural logs are incredibly common very very very common so now let's go backwards we've practiced going from exponential notation into logarithmic notation now we're going to go backwards let's practice if if a logarithm takes your exponential and separates the base from the exponent a logarithm should put it back together so when we identify this one important thing is to identify what your what your base is and what your exponent is so let me take a look at it if this is a logarithm my base is three and my exponent because it's a logarithm is negative two your base in your exponent are separated in a logarithm your exponential should put that back together now what this says kind of kind of strange it says in order to go from a base of 3 to 1 9 i would need a negative 2 as in my power on the 3. it says it backwards because it's an inverse your exponential should put that back together for you should say your base and your exponent get put back together on the same side of the equation and should equal your argument so 3 to the negative 2 equals 1 9 that actually makes sense 3 to the negative 2 power is 1 over 3 squared that is 1 9. so moving back and forth is really important for us we're going to start seeing that how about log base b of four equals two a logarithm has separated my base for my exponent so that's my base that's my exponent and exponential will put it back together now can you go further could you solve that yes you could but be a little bit careful when you're talking about logarithms your bases have to be positive they came from exponentials didn't they one of the key things here was that the base of an exponential had to be positive so if we ever see a base that is negative that is disallowed we can't have a negative inside of our our base just like we can't have a negative inside of our argument so keep that in mind all right you should probably try the next few of them just putting these putting these in exponential notation be careful on these two because they are common log and a natural log counter respectively here we have a logarithm logarithms have always set already separated the base from the exponents so if we identify those we've got a base of 2 we have an exponent of x an exponential will put that back together and give you 2 to the x equals 6. these last two there are the same things they they are just logarithms what i want you to notice especially on this one is that that argument's unsolved and your exponential will solve a logarithm for you if you need to solve it now this is already solved i mean this we really shouldn't be doing exponential notation but we're just practicing this i want you to notice that your exponential will solve that for you so let's go back to it we have ln 4 equals x is it a logarithm yeah of course a logarithm has a base yeah it's base is e we don't write it we'll never do that again but its base is e there so what the logarithm is done is separate the base from the exponent and exponential will rewrite that it will group your base to your exponent and equal whatever your argument is it puts it in a different form it says something different it says a base of e needs or what do you need in order to go from e to four as an exponent well you would need i don't know x this says e to the i don't know x equals four that's what that says so log of x equals 3 that implies a common log that implies a base 10. we don't write the base 10 but it does have a base in order to go from a logarithm to an exponential you're going to regroup your base from the exponent which has been separated so our base is 10 our exponent is 3 and that would equal x i asked you to know something i asked you to notice that this logarithm had a unknown argument this exponential solve for that so the the two things we're grasping one right now is firstly how to go back and forth from logarithms to exponential notation and exponential notation logarithms but the second thing i want you to be grasping here is that a logarithm will solve an exponential like that and an exponential will solve a logarithm why well because they're inverses and they have to that's what inverses do inverses undo their root functions so a logarithm has to solve an exponential an exponential has to solve a logarithm just like addition has to solve subtraction subtraction has to solve addition they work together to undo each other they work together even compositions already see it seem really cool when you compose a logarithm and exponential they're also going to undo each other we're going to start seeing that so come back with just uh one more way for you to kind of think through these before we get to graphing i hope that you don't breeze through this part of the video i hope that you really spent some time on it here's why this is going to start down our road to solving logarithms with exponentials it also will sort of fill in how to solve exponential logarithms but mostly how to solve logarithms with exponentials so how how in the world does it does it do that well it goes both ways actually but here's here's one thought on it can you think through what a logarithm means well enough to determine what these things equal i'm going to give you two ways to go about and do it the first way is really using some thought of what log of the masks these ask questions like if you have a log base 5 of 25 here's what it's asking you what power do you need to go from 5 to 25 remember this in a logarithm is always an exponent this is your base of some exponential this is always your exponent so this asks if i'm missing this what power do i need to go from 5 to 25 what i would need a power 2 and this is going to be a power 2. now what if your brain doesn't work that way if you go i what power do you need to go from one what power do you need to go from one third to nine my brain does not work that way how in the world am i gonna do that an exponential will solve it and i'll show you if you call that x and simply do exactly what i showed you the last part of this video where it said all right if that's a logarithm can i write this as an exponential yeah i would take base to exponent equals whatever my argument is but wait a second that looks a lot like an exponential where we can find i just do the math in my head where we can find common bases and if your bases are the same your exponents also must be the same now this one might have been pretty easy to do in our head but this one doesn't really seem that way now i can tell you it's negative 2 but it might be easier for you to do this and say if that's my base and then that has to be my exponent if i write these as common bases you know wait a minute common bases how do i get a common base with one third do you remember common bases do you remember that one-third whenever you see a fraction write it with a negative exponent nine you can write as three to the second power one-third you can write as 3 to the negative 1 power well that will lead you to 3 to the negative x equals 3 to the second as soon as you have common bases your exponents have to be equal and x equals negative 2 and that is true 1 3 to the negative 2 power would give you reciprocate and square it 3 to the second power which is 9. that's true statement how about this one how about this uh this common law log all that's got a base of 10 log of base 10 to the square root of 10 so what power do you need to go from 10 to the square root of 10 or 10 to what power gives you square root of 10. do you remember that a square root is a fraction of one half as an exponent so this is going to be one half 10 to the one half power would give you the square root of 10. could you do it a different way could you write this as an exponential and think oh my base is 10 this is separate in my base from my exponent if i know that 10 to the x power gives me an argument square root of 10 and i remember that square roots are fractional exponents of power over root then my bases are the same and x would have to equal one half the same sort of things are going to work here we're going to run through them really quickly try them in your head though before you start doing this technique try them in your head just think through it can you think 5 to what power gives you the cube root of 25 it might be valuable to think of 25 as 5 squared because 5 to the 2 3 power would give you power over root that'd be 5 squared inside of a cube root if that's not working for you yeah sure call this x an exponential will at least make it easier to think of these most of the time and especially if you have common bases base to the exponent remember longer than several bases from exponent exponential puts it back together 5 x equals 25 to the one-third power cube root is a one-third power if you need to find common bases then you need to write 25 as 5 squared and then we know up power to another power we multiply those and we can see that as soon as we have common bases our exponents are equal and that's a true statement 5 to the 2 3 power would give you 5 squared inside of a cube root that's five squared inside the cube root [Music] this one's really important i need to do on your own uh these two i'll walk through with you this one's really easy it leads us to to the idea of compositions but this one's kind of hard square root of three what power do you need to go from the square root of three to nine so basically you need to undo the square root and then square it again so undoing the square root of power two that would give you three then another power two would give you nine so two to the the second power to a second power that's a power four this answer is going to be four so we've thought through it if you would like you can write this as well logarithm separated base from power that is not right logarithms have separated the base which is a square root of 3 from the exponent which is x and puts it equal to your argument that's the exponential notation for this logarithmic expression now we'd have to find common bases so change your radicals let's make this three to the one half power cannot lose that x most of the time i write that first before i even deal with this and then 9 is 3 squared but if we do that then 3 to the one half to the x we know that we can multiply those exponents so 3 to the x over 2 equals 3 squared our bases are the same our exponents therefore must be equal and x has to equal 4 by multiplying by 2. all right now what about this one i told you that this is really a composition i really really want you to see it whenever you compose whenever you compose two inverses with the same base in the case of logarithms exponentials they will cancel out i'm going to prove it to you that i'm going to show it to you so firstly it's it's an absurd statement which is why not absurd um it's a redundant statement or a trivial statement maybe it's the best word which is why when you see the composition of an exponential logarithm they cancel out think about it this is a logarithm yeah it's got a base of e so if i wanted some really poor notation i'd write the base of e but wait a second this argument also has a base of e and it happens to be an exponential so we have composed check it out we've composed an exponential onto a logarithm with the same base whenever you do that or a logarithm onto an exponential the same base it doesn't matter whenever you do that these things are going to cancel you are going to get 3. because here's the statement this is asking e to what power gives me e to the third wait a minute e to what power gives you e to the third you just said it had a third power yeah e to the third power gives you e to the third power it's trivial because you've composed a function with this inverse and they're going to cancel they're asking kind of you to complete something that's already been completed and it's going to give you three now if you want to see this a little bit more detail you can do it this way you could say e to the x power equals our argument which is e to the third here's our base here's our exponent put them together equals our argument but can you see that's already completed common basis for you it's already done you're gonna get x equals three any time you see this an exponential inside of a logarithm of the same base those are going to cancel anytime you see an exponential with a logarithm inside with the same base in in the exponent place those are going to cancel also we're going to get those in properties or logarithms but i wanted you to see it here the last one before you can finally get on to some graphing and some domain because we're getting very used to how a logarithmic structure will be able to identify what the base is and we'll be able to identify what the argument is we need to know this one so log base 2 of 1 equals what let's look at it let's call that x what an exponential would do is take your base put it back together with your exponent and equal it to your argument that's what exponential notation does you should be very good at it right now going from logarithms to exponentials we already practiced a lot of going from exponentials to logarithms so now we go right oh wait a minute uh what does x need to be what what does x absolutely have to be to take two to some power and get one there's only one thing we can do there in fact this is one of the key points of our exponentials isn't it because we said every single exponential ever is going to have one two to what power gives you one translates to every exponential has a point zero one on it why because we know that anytime we take a base that's a constant to an exponent that's zero base to zero is going to give you one any exponent to the zero power is going to give you one that translates to this please get this if any exponent to the zero power is going to give you one any logarithm of one is going to give you a zero every single time this is why no matter what the base was we had a key point on our logarithms of one comma zero do you remember that we'd always have one comma zero for our logarithms it was a key point for us why if an exponential gives you zero one because you plug in zero you get one no matter what whatever base you have then any logarithm is going to give you one zero because for any logarithm you plug in one you're going to get that that zero it says your base raised to a zero power is going to give you one no matter what no matter what base you have two so what if that wasn't a base two if it was a base three or square root of three or five or i don't care what this is this is going to be true that's why 0 1 is on every exponential graph and one zeros on every logarithm i hope that makes sense to you this is going to be one of our properties of logarithms as well when we get there but i wanted you to understand that because it solidifies one of our key points of one zero for every logarithm regardless of what your base is let's come back and we'll do some domain and then we'll talk about some graphs very quickly okay let's learn about some domains of logarithms there's one thing we learned we took a look at the graph the beginning this video that said if your range of your exponentials from zero to infinity so basically just positive then your domain for logarithms has to be the same thing range became domain for logarithms what that means is that the domain of your logarithms state that your argument the inside of your logarithms must always be positive not zero but positive think square roots but not equal to zero so we have to have something that is strictly positive in this there's there's no no way around that and so when we go ahead and we find domain for our logarithms it's actually a really simple task it also means that we include one more thing in our knowledge of domain so up until this point you really only had two features that you had to worry about one was denominators you could not have a zero on the bottom of your fraction number two was square roots you had to have the argument of your square root greater than or equal to zero and now we have logarithms i don't care whether you have an ln or a common log natural log common log or a log with any other base it doesn't really matter for every single base of every single logarithm ever the argument must be greater than zero that's all there is to it so whenever we see a function that has a logarithm in it we're going to look at its argument the stuff inside of parentheses the seventh century longer than say that because this is based on some logarithm with i don't care what the base is that argument must be greater than zero so we take a look at it go okay is that a logarithm yeah it's going to have some domain issues all right what are those going to be let's take a look at our argument and say this is a logarithm of the base of e i know that the inside of it must be greater than zero so what that says is i can only allow inputs that make that argument strictly positive i can only allow inputs that keep this thing greater than zero that's the argument greater than zero that's what we mean here the argument of your log must be positive okay if we add three that says you can have any number in the world as long as it's bigger than three anything less than or equal to that is a problem because it would create something undefined for your logarithm logarithms do not allow arguments that are negative logarithms that are not shifted this is shifted oh man do you get it this is shifted three units to the right this is inside your logarithm that's a minus three minus these two the right shift so before shifting it logarithms did not allow zero or negatives when we start shifting the logarithm we start shifting our domain a bit but it still bears out that our argument must be positive that says you can only allow values that are more than three if you try to plug in three you get ln of zero why don't you try that right now put in your character there's ln zero it's going to go error you're killing me why don't you understand i can't do that that's a little melodramatic but you can't do that it's going to give you now i promise try it right now press ln 0 enter and you're going to go sorry all right so this is your domain your domain is you gotta have numbers bigger than three anything besides that and you're you're blowing this thing up uh you can't allow arguments that are zero or negative let's move on so we have another logarithm it looks really nasty um we're going to get into the graphing stuff that looks similar to this so why in the world well maybe there's some things we can identify here like that's a shift up three this is written out of order that's a reflection that's a vertical stretch this is gonna have some sort of a shift to the right of some amount but that's not what we're talking about yet we are going to get there but right now we're just talking domain so we'd say this is a logarithm it has a base of 4 this is my argument and i know that in logarithms i need to plug in values that keep my argument positive here's what not to do do not set this equal to zero and try to solve it that shows your instructor whoever that is that you really don't understand the concept and it also is going to lead you to make an error with your inequality that we have right now because you're about to multiply by negative 2 or multiply by 2 and divide by negative 1 either way you're going to have to reverse the inequality so you look logarithm i know the argument must be positive so only allow inputs that make that happen let's subtract five you can multiply by two that would not change your inequality but dividing by negative 1 or multiplying by negative 2 here as soon as you do that that reverses your inequality says you will need to have values of x that are less than 10 in order to satisfy the fact that that needs to have an argument that's above zero not 10 itself try to plug in 10. if i plug in 10 you're getting 5 minus 5 and that's zero that's a problem plug in 11. that actually plugged in 12 it's easier plug in 12 something bigger than 10. if i do that 10 12 divided by 2 is 6 5 minus 6 is negative 1. your logarithm is not going to be happy with that it's going to go no i can't have negatives in there so keep it positive that's why we have to have values that are less than 10. if you forget to do that it's going to give you completely the wrong domain also notice one thing here please get this that it's not about plugging in negatives it's about plugging in values that keep this whole thing positive so in this domain you have negative values if you plug in negative 20 you can do that negative 20 divided by 2 is negative 10 but 5 minus negative 10 is 15. that's perfectly fine that's absolutely okay so it's it's more about this keep the inside and make find the inputs find the domain find the interval of x-axis that keeps your argument positive and that's what this is doing for you last one we're going to bring something back here we have a logarithm it's got a base of five it's got this argument we all understand hopefully at this point that your logarithm has to have an inside an argument that is always positive the question is do you remember rational inequalities that we did a few videos ago oh man that stuff's coming back yeah it's coming back because we can have fraction side logarithms which means according to domain that you need to solve for the interval on which that's defined wait a second so now we're using something we learned and something that we're learning yeah that's the way math works so if we have x plus one over x needs to be positive needs to be greater than zero do you remember what to do here do you remember that we create a number line that we identify what our problems are x cannot equal zero due to the fact that this is a denominator and if we allow our denominators equals zero and we cannot cancel the factor that this is a vertical asymptote and it's odd due to the fact that has a power one this should be just going wow oh my gosh i remember how to do that on our numerator we'd set that equal to zero this would give us x-intercepts or in general where this function intersects this function that is an x-intercept and that is a cross we can put this on a number line and we can use our vertical asymptote and our x-intercepts to determine whether or not we actually have sign changes uh due to the fact that they're odd or even multiplicities man i hope that makes sense so put these in order we can test one point something like x equals one and just see if it's true or not if we evaluate x equals one then we can say one plus one is two two over one is two two is bigger than zero that's a true statement what that means is that every value on this part of our x-axis every single value is going to create a true statement here every single value is going to create this thing being positive why is it important because if we have values that create this thing being positive this thing is your argument this thing is saying these numbers are allowed in that function this is defining your domain you've just found the values of x that keep this positive and that's what we're looking for for logarithms so this is one part of that interval just remember that when you have odd multiplicity your truth changes when you're doing what's called a sign analysis test or checking your intervals so true here means false here false here across means true here and so your intervals that cause this to work just like that negative infinity to negative one and then zero to one i'm sorry zero to infinity um i said one here and that's a very common mistake sometimes when we check a value we accidentally either put that value in our interval or we make this part of an interval that doesn't really need to be there so our working intervals the things that keep our argument true that satisfy this rational equality which represents our argument are any value from negative infinity to negative one those values are all going to create a positive argument and are therefore allowed in that logarithm that's what domain means along with zero to infinity any x value in that interval is going to cause this to be positive and therefore is allowed in our logarithm you can try some if you want try something like 10 10 plus 1 over 10 is positive try something like negative 2 if you'd like negative 2 plus 1 is negative 1 over negative 2 well that's negative that's positive 1 half negative 1 over negative two is positive one half or just one i don't care if it's fraction we're just want to keep that positive and you can do that all day long but these are the only two working intervals that we have for that so that defines our domain i hope that i've really made this make sense to you so far i hope that from the graphs we understood that logarithms need positive domain need values that keep your argument positive now sometimes you can allow negatives but that's only because when you plug them in your argument becomes positive that's what we're looking for so we're going to come back in another video and we're going to start graphing logarithms with some transformations so i'll see you for that you