[Music] you [Music] [Music] [Music] [Music] you [Music] [Music] [Music] welcome back to lockdown math today we are going to be talking about logarithms and kind of a back to the basic sort of lesson the intent is for someone who maybe is not familiar with logarithms yet or who has seen some of the rules but has been confused by them and as always to kick things off I just want to get a sense of where the audience is at right now because I have a couple suspicions but I think doing a live poll to see where everyone is might be helpful so if you can go to 3 B 1 B Co /live which you can find the link in the description and you can also just follow the link that's up on screen here you'll find a poll and at the moment it asks what best describes your relationship with logarithms a I've never heard about them before we're never learned about them before B I've learned about them but sometimes get confused by all of the properties see I understand them but wouldn't know how to teach them and D I understand them well and could comfortably teach them to someone else to make them understand well too so we've got a good split this is good and interesting like I said the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say oh here's a place that you can go for how I think you know how I think you could approach it intuitively and for those who are in the camps of let's say understanding it but not knowing how to teach it I think there's an interesting meta question around logarithms because I was scrolling around a couple teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know so if we skip ahead of that where we're gonna go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have you know sort of nightmarish recollections of what high school math was like and what logarithms did for them it's often those particular formulas coming to mind and what I want to do today is try to talk through one how to think about them but also just on the meta level of if you're teaching somebody algebra what are the points worth emphasizing what's the way to get it built in their intuitions so let's go ahead and see how people are answering on this one we've got a pretty even split between three three possible categories all right so the most common viewer right now answered see that they understand them but they wouldn't know how to teach them the second most common are those who said they understand them well and they could comfortably teach them so this is sort of revealing that okay the existing audience on the channel at the moment that's people who are into math and they're going to be pretty comfortable with logarithms and then after that be people who are confused by some of the properties and then at the bottom people have never learned about them now if you watching are one of the people who answered a that you've never learned about them or that you're not comfortable keep in mind this lecture is actually meant for you so when we're doing some live quizzing and we're gonna have an answer I'm here and we're gonna have people you know answering probably pretty quickly probably correctly given that as you can see most of them are coming in comfortable with logarithms don't let that be intimidating don't let that be something that indicates you should be answering quickly or you should necessarily be getting the right answer right so I think the most intuitive way to think about logs if you're just going to start off is to say at a very high level logarithms are something that can take in a power of 10 and it just spits out the number of zeros at the end of the number it does more than that as we'll talk about but just at the very high level this is getting a sense for what they do and I think it's pretty important because what you're saying here is hey look as we walk from one to ten to a hundred to a thousand to ten thousand each step were multiplying by ten the output of the log the log of that value just steps up in these increments of one so you can think about this as saying hmm when I plug in a number to the log if that number happens to be a power of 10 I'm just counting the number of zeros what's log of a thousand oh it has three zeros on it what's log of a million I would six zeros on it it's the zero counting function and I think you can get yourself very far in terms of understanding very various properties of logarithms just by thinking in the back of your mind okay when I plug in a power of 10 it's a zero counting function i plug in a billion how many zeros are at the end of a billion and these numbers are all very intuitive to us we see them in the news we we anchor our intuitions in terms of how many digits does a number have and logarithms are just capturing that idea and in fact logarithms have been a particular interest in the last couple of months what's kind of interesting is if you look at this Google Trends chart that's just looking for logarithmic scale okay people searching for the term logarithmic scale you'll notice you know there's kind of an ambient interest sitting around in September of 2019 up to forever of 2020 and for some reason suddenly there is a strong interest in understanding logarithmic scales from the general public I wonder why that is and we see this sitting around early March or so and of course this is because this is when the corona outrace outbreak was really starting to kick into high gear and everyone wanted to understand exponential growth and a common way that exponential growth is plotted is with what's known as a logarithmic scale so I actually made a video about this and in it I was you know creating some animations and wanted to illustrate this idea of exponential growth and the main idea here I'll go ahead and skip back to a different animation is if you're tracking the numbers in this case this was the number of recorded cases of kovat 19 outside of mainland China in the months leading up to March you could just track what the absolute number is but the pattern that you'll find is that as you go from one day to the next you tend to be increasing multiplicatively it's a little bit like earlier we were seeing the powers of 10 one step to the next you're multiplying by some amount the way that the virus was growing was very similar from one day to the next you're multiplying not quite by a constant but in this case for this sequence of days it was around 1.2 in that region you're multiplying by something so when you're plotting this it ends up looking like this you know classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the overall pattern is so a common trick is to say instead of looking at this y-axis that increases linearly as in here I'm going from 5k to 10k 10k to 15 K 15 K to 20 K each step is additive we're adding five thousand instead use a y-axis where each step is multiplicative so you're going from 10 to 100 100 to 1000 1000 to 10 10 thousand all of these are increases by multiplying by 10 and what you could say is the y-axis is now plotting not the total number of cases but the logarithm of the total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend would do and you know it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases of something like this that is what it is so I kind of fast forward in the animation I made for that video and what's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is that line going to cross a million which because the y-axis is growing but with multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you're understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively just drew out that line and in fact fast forward to around April 5th which is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line that starts to taper off but point being anytime that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined okay and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so max 182 asks us additions inverse is subtraction multiplication inverse is division but I never truly understood if exponentiations inverse was n through ting or logarithms if either one could be you know if either one could really be called that way can they that is such a fantastic question max and I think it comes down to the fact that with addition and multiplication you're not or I'll just draw it out for you actually this is gonna be easiest if we have some kind of exponential relationship let's say 10 to the power 3 is equal to 1000 there's three different numbers at play here we're showing a relationship between the 10 the 3 and the 1,000 and aside from writing it with an exponent there's two other ways we could write that same relationship we could also say that the cube root of 1,000 is equal to 10 this is asking a thousand what number raised to the third is equal to one thousand that's sort of what the cube root is asking and another way that you could raise the exact same thing is to say the log base 10 of 1000 is equal to three okay three different notations the same exact relationship a while ago I made this video about an alternate possible notation that would Center in the idea that you think of this relationship between the three numbers with a triangle where you'd have our 10 sitting down here the power sitting at the top 10 to the third and the thing they equal sitting to the lower right and whenever you want to talk about a function or an operation between two of these numbers you indicate which one of them you're leaving out so to write 10 cubed we would include the 10 on the lower left the 3 on the upper right and then we leave out that bottom one so this is indicating that we're taking a power 10 10 cubed for that radical what you would say is we know what's on the bottom right something to some power equals a thousand we even know what the power is so than cubed equals that thousand but the thing we don't know is on the bottom left that is the sense in which radicals are an opposite of exponentiation where if the thing you don't know instead of being on that bottom right is on the bottom left but what logarithms are it's an inverse in another sense because what it's saying is in this triangle relationship we know the base it's ten we know the power that it should be a thousand but the thing we don't know is what's in that exponent so to answer your question maybe even more briefly we could say that 10 to the X the inverse of that the inverse is the log base 10 you know of Y of some other variable whereas if you're taking X to the tenth some unknown raised to a power the inverse of that is going to be the tenth root of some other value and on the triangle it's basically asking which of the things do we consider to be a variable so are you considering that lower right to be a variable quantity are you considering that top to be a variable quantity and what what is the unknown but I really liked this idea of making explicit how we have three totally different notations for the same exact fact one of them you're using relative positions of the numbers one of them we introduced a new symbol this radical and one of them we introduced a new word log so these three syntactically different ways to communicate the same idea seemed wrong and so I made this video about you know an alternate possible notation and while I don't necessarily think that oh we should teach logarithms with this triangle because convention is what it is so it's better to start getting people used to the usual expression what I do like about it and starting off with it is when you see and think about this triangle it's really emphasizing that what the log wants to be is that exponent every time that you see log of some value you should think in your mind okay whatever this number is it really wants to be an exponent it wants to be an exponent and we'll see more of what that means as we go on okay so every time you see a log it wants to be an exponent this value three and more specifically it should be an exponent sitting on top of whatever that base is now in terms of convention for the first part of this video I'm just going to be using log without a base written on it to be the shorthand for log base 10 because log base 10 will be the most intuitive thing out there you should know that often in math the convention instead is that log without anything might mean log base E there's also another notation for that Ln for natural log we're going to talk all about the natural log next time so don't worry too much about that right now and there's also yet another convention often if you're in a computer science setting log without any added sugar to indicate what it is defaults to meaning log base two so this can sometimes be a source of confusion but it basically depends on what discipline you're in in math not math math people really like a base of e we'll see why next lecture in I don't all say engineering but really it's anything where you want good intuition with our normal base ten number system log means log base ten and if you're curious often in computer science settings log base two comes up all the time so like I said in the back of your mind if you're trying to think of some of these properties just resting on the idea that log counts the number of zeros at the end of a number that can get you a really far away so we're going to start going through a couple of these properties and I want to do this just with a set of practiced examples so we'll transition away from the pole and this time to the first proper question and the question asks you which of the following is true a the log of a thousand times s cubed C log of a thousand times x equals three plus the log of X D log of a thousand times x equals three to the power of log of X and E none of the above okay and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately they're going to be answering correctly but if you're someone who doesn't don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of ten and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that [Music] [Music] great okay so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of a thousand times X is the same as taking three plus the log of X okay and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of a thousand times X well I don't know let's just plug in something for X log of a thousand times a hundred okay well we know how many zeros are going to be in the final answer here well a thousand times 100 is a hundred thousand we already intuitively have this idea that when we multiply two powers of 10 we're just taking the zeros the three zeros from that thousand the two zeros from that hundred and we're putting them next to each other so it should be five total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the three zeros from that thousand plus the two zeros from what that 100 which we could also write by saying the number of zeros in a thousand plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of ten that's just communicating what's already a super intuitive idea for a lot of us if you take two powers of ten and you multiply them you just take all of their zeros and kind of cram them on to each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of a times B it equals the log of a plus the log of B and now anytime you see one of these logarithm rules if you find yourself you know squinting your eyes or a little bit I'm confused by how to remember it just plug in examples I'm being redundant I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you our okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case saying log of a times B and breaking it apart we could just think oh that log of a hundred times a thousand which is five there's five zeros in it breaks up in terms of the number of zeros in each given part great wonderful so carrying that intuition further let's try another practice problem and again if you know it great you'll be able to answer it fine but maybe think not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is and so there's two potential audience members there are those who are interested in the lesson itself and then those who are interested in the meta lesson so a question asks again which of the following is true a log of X to the N is equal to n times log of X B log of X to the N is equal to log of X to the power n C log of X to the N is equal to n plus log of X or D log of X to the N is equal to log of n times log of X or none of the above so again take a moment to answer as people do answer we'll start to see them roll in and if you are struggling to think of where to even start just plug in some powers of 10 and see what intuition you might get from it [Music] you [Music] you all right so it looks like just about everybody is landing on one particular answer hopefully it's the right answer like I said all to be expected so let's go ahead and see if our majority ended up correct and they did so the correct answer here is a which it looks like 4000 of you got congratulations telling us that log of X to the power n is equal to n times log of X so again let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case for log of X to the power n let's just try it with a hundred to the power three and you can try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because a hundred cubed we can think of that as taking well that's three copies of 100 I'm taking three copies of 100 and when I multiply all that out and I think of log is counting the number of zeros when we say oh it's going to be some number that just has six zeros on it that's what it means to take a hundred times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be six but if we think actually why was it six not just that's the number of zeros inside the million where that six came from is that we had three copies of that 100 and each of those 100 had two different zeros see that way it's a more general way you can think about it where if instead of taking a hundred cubed we were looking at a thousand cubed or a thousand to the N or X to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well let's see it's not x times the number of zeros they were in whatever we substituted for X which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that thousand ten thousand counting the number of zeros in each one of them we for so it would be n times four and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it every statement of the definition if I'm taking log and I'll just reinforce eyes its base 10 of 10 to the power n we can kind of think of that little n is hopping down in front and becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the N oh well if the answer is n okay now with every given logarithm property that we have so in this case we just found one log of X to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the end that hops in front this corresponds to the exponential property that if I take 10 to the X and raise that whole thing to the power n that's the same as taking 10 to the N times X and this corner just gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the things sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the X the the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the what's the exponent of the ten so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see a exponential the entire 10 to the X expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs okay and we saw this above right the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn Exponential's inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log B is playing the role of the X and the y in the expression on the right okay so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true log base a of B is negative log base B of a okay log base a of B is one divided by log base B of a log base a of B is one minus log base B of a log base B of A is log base a of 1 divided by B or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that [Music] [Music] okay so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case the correct answer of the choices we have comes out to be be that the log base a of B involves taking one divided by the log base B of a again let's think this through both in terms of an example and then in terms of a more poofy systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like a thousand it's counting the number of zeros we get three so let's try swapping the bases and see what this should mean log base a thousand of 10 okay well what is this asking right maybe you can't think of drawing the little triangle saying something like we know a thousand to something is equal to ten a thousand to the what equals ten well if 10 cubed is a thousand that is the same thing as saying ten is equal to a thousand raised to the one-third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of a thousand it's 1 divided by the log base 10 of a thousand so more generally you might guess based on the single example that when we swap the base with what's on the inside it corresponding it corresponds to taking 1 divided by what's on the outside there and again you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and Exponential's wonderful so again let's hide where some other thing and the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having a pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around and so what we just found log base B of a if you swap those it's the same as dividing by one what this corresponds to often exponential land is if you take B to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals B again it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base B of A is playing the role of that X okay and the expression log base a of B is playing the role of whatever sits on top of the a okay and then symmetrically the whole expression B to the power X is playing the role of the inside on the left it plays the role of the a and the whole expression a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next from one step to the next things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real-world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms okay one last one before we transition to a little bit of a real-world example get rid of what we had here [Music] and now which of the following is true log of a plus B is the same as log of a plus log of B log of a plus B is equal to log of a times log of B log of a plus B is equal to 1 divided by log of a plus log of B or log of a plus B is equal to 1 divided by log of a times log of B or none of the above ah and now we don't have as much consensus do we very interesting we've got a horse race between two so I will give you a moment to think this through [Music] [Music] while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 where if you're going from 1 to 3 to 9 to 27 to 81 all of these we can say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1 3 to the what equals 1 they answer a 0 in general the log of 1 no matter the base will be 0 log base 3 of 3 3 to the 1 equals 3 is 1 similarly log base 3 of 9 is 2 you might wonder what my question is but it'll help to draw all of these out and for my own pleasure here let me just write out one more log base 3 of 81 is 4 now I've heard that ostensively if you ask a child's let's say around like five or six years old what number is halfway between one and nine and you say what number is halfway that their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2 3 4 5 6 7 8 and if you go right halfway in between I'll land on 5 but if you're thinking in terms of multiplicative growth where to get from 1 to 9 it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with if you have anthropologists studying societies that haven't developed counting systems and writing in the same way that modern societies have they'll enter 3 for this so my question for the audience if any of you watching right now have access to a small child let's say in the range of 5 years old see if you can go ask them what number is halfway between 1 and 9 and if you can let us know on Twitter what the what the child says what their actual answer is because I I don't know why I'm a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great okay so 2,400 have you correctly answered that it's none of the above okay the log of a plus B doesn't satisfy any of these nice properties and in general unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness plugging some powers of 10 all right if I ask you log of a plus B what you might start thinking is okay let me just plug in some examples like ten thousand and one hundred and I ask myself if I do this zero counting function of what's in that input how many zeros are in it but it's weird because when we add 10,000 100 well we're no longer at a clean power of 10 and okay that's fine you know often you're taking logarithms of things that aren't clean powers of 10 but it becomes very strange to ask how you express this in terms of log of 100 which was 2 + log of 10,000 which was 4 because if you look at log of 10100 it's asking 10 to the what is equal to 10,000 100 you might say I don't know it's gonna be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is gonna be something that's kind of like the log of 10,000 but that just feels like a coincidence based on the two numbers that we happen to put in there's not a nice systematic reason coming there so maybe you guess oh the numbers a and B are very different it's kind of close to whatever the maximum of them is but it's very bizarre and most importantly for the sake of the quiz if you just look at the options that it's giving you if you try this out with any particular numbers you'll find that none of those actually work so all is good sometimes you get something that looks like it's gonna be a nice property but it doesn't end up being a nice property and I also think that's important rather than just finding yourself only working with the various you know log of a times B or log of X to the power n these things that have a nice rule sometimes you're out in the mathematical wild you're working on some problem you have a logarithm expression and it's adding things in the input and you want to be able to have familiarity with the fact that that's kind of weird that you're not going to be able to simplify but if you you know if you hadn't thought about that before you might wonder oh is there just some formula that I haven't seen before so with all of that let me go ahead and take a couple questions from the audience before we transition to a different sort of example so it looks like uma Sherma asks can can the bass be zero that's an interesting question okay can the base of a logarithm be zero well in terms of our triangle we might think of that as saying you know zero to some kind of power X is equal to some other value Y all right this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x 0 to the what equals x now the issue here is that 0 to anything ends up being zero right so if we're just going to be thinking of log base zero of Y for any other input Y you know you want to input something like 1 or 2 or pi anything you might want you're asking the question zero to the what is equal to 1 or 2 or PI or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes log of 0 it's a perfectly valid function it's only defined on the input 0 but even then you'd have trouble trying to finagle what you want there because saying 0 to the y equals 0 it's like anything anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base 0 is entirely 0 it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other so that's a great question can you have a log base 0 now back to the idea of where these things come up in the real world one one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is okay and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 okay and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms will be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much okay which is a small a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two up to the nuclear age so that is noteworthy and the first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us and evidently a single step up from four to five corresponds to going from 1 kiloton to 32 kilotons and that was everything that was a 4.0 versus an earthquake there was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next and going from 2 to 4 was evidently multiplying by about a thousand okay and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how how much does this grow partly because of the the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but raining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one Megaton which we might think of as not small atom bomb Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one Megaton that is evidently the magnitude of the double screen flat earthquake in Nevada USA in 1994 I didn't know what that was thanks for capilla in terms of frequencies by the way I also looked these up evidently ones that are less than two those happen all the time there's like 8,000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and for those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew but as we get even more intense into this five and six range which you know we're well above the atom bomb scale now we're only merely at around two per day and you know I'm sure that a geologist can come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the Earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some some spot like a city where lots of people live now just verifying that our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us a lots more examples in between maybe giving the illusion that that's a bigger step that it actually is and indeed that's the difference between one Megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the Cold War the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100-megaton bomb but talked themselves down from that 50 megatons we're talking start off at that 30 to kilotons of the Nagasaki bomb multiplied by 32 to get a Megaton multiplied by another 32 rights we're talking about a thousand times the strength of the world war two ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0 it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what looked like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so it I mean when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tempt to come about it's um when humans want to create a scale for something that accounts vary hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the earth the size of a lady right and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically puts squishes those numbers between zero and ten you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of ten decibels that corresponds to multiplying by ten so rather than a step of one multiplying by ten it's a step of ten that multiplies by ten so that kind of makes the math of it a little bit screwy but the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels it's a lot quieter in terms of the energy being transmitted and going from you know what would it be 60 to 70 or 70 to 80 those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the corona virus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by one you're multiplying by 32 well we could think in terms of a log with base 32 I could say if I take the log of I'm just going to call R the number for the Richter scale I might think of this as log base 32 and that's going to correspond to no no no I'm doing this wrong no that's not the thing that's logged we take the log base 32 of the big number of the the T&P number something that was like you know one Giga ton or one Megaton it's 1 million tons the log base 32 that should correspond to the Richter scale number but there might be some kind of offset so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking no 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see as you increase that are step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is all right now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log if you're able to compute one type of log whether that's base 10 base to base e you can compute pretty much anything else that you want okay now to get our intuitions in that direction let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know this is this is a half way reasonable question this should be nice this is just gonna get us prepared to translate from base to contexts to base ten context and it's also a good intuition for understanding powers of two to have in general the relationship that it has with powers of ten because it's this lovely kind of coincidence of nature that these two sort of well you'll you'll see what I mean they play nicely with each other so a question asks given the fact that two to the tenth is 1024 1024 which is approximately 1000 okay so you can if you're being a little bit loose with your numbers and you're just making approximations to to the tenth basically a thousand which of the following is closest to being true log base two of 10 is approximately zero point three log base 2 of 10 is approximately sorry log base 10 of 2 is approximately zero point three log base 2 of 10 is approximately one third or log base 10 of 2 is approximately one third in which of these is closest to being true based on the fact that 2 to the 10th is essentially a thousand I'll give you a little moment for that interesting that we've got kind of a split on this one so I'm wondering if they're gonna be numerically pretty similar or if they're gonna be conceptually similar or if there's even a difference between those two [Music] so since answers keep rolling in I'm gonna give this give us a little bit more time so anyone at home watching hopefully you already have a pencil and paper out to be noodling through these yourself that is the spirit of the lectures that we're doing if you don't now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're gonna build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen give you a little bit more time here as the answers seem to continue rolling in [Music] you [Music] okay so I'll go ahead and grade it now and let's see how people did on this one so the correct answer is B which is that the log base 10 of 2 is approximately 0.3 and it looks like 1850 of you correctly got that so congratulations but the close contender not at all a unanimous decision here looks like it was d which is that the log base 10 of 2 is around 1/3 so that's good they're very numerically similar right that it's either 0.3 or around 1/3 which is 0.33 3 3 repeating but the question was asking which one is closest to being true and let's see how we can think about this so it points out that you have a power of 2 which is a thousand 24 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other so what does this mean if log base 2 of 10 is equal to X that's the same thing as saying 2 to the X is equal to 10 right it's asking us 2 to the what equals 10 so what we have here is an expression 10 cubed is approximately equal to 2 to the 10 so what I might write out is we know that 2 to the 10th instead of writing it as a 10 I'm going to write that 10 as 2 to the X where X is the number such that 2 to the X is approximately is equal to 10 so if that cubed is the same as 2 to the 10th this is I'll just write out the full details the same as saying 2 to the 3x is equal to 2 to the 10th and exponentiation is a nice one-to-one function so it's okay to just say whatever's going on in the input if the outputs are the same the inputs must also be the same you can't do that with every function people seem to think you can do that with any function but you just can't and what that means is that X is about X is about 10 thirds okay which good so place two of ten is about ten thirds so if we looked at our answers though that's not actually any of the options we've got various things asking log base two of ten being around 0.3 or one-third so it looks like instead we should try to reexpress this as log base 10 of 2 and well-enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there so log base 10 of 2 is 1 divided by this amount which is 3/10 which is 0.3 great so this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of two so if I ask what is the log base two of a thousand like we just saw it's approximately the case that 2 to the power 10 is equal to a thousand and because we're doing things in logs I'm just gonna be writing it in that way a log two of a thousand is approximately 10 similarly log base 2 of a million well let's see if we have to multiply two by itself about ten times to get to a thousand we should have to multiply by itself around 20 times to get up to a million and indeed log base two of a million is approximately 20 it's a little bit smaller but this is kind of a nice approximation to have in your mind and then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion saying how many times do I have to multiply two by itself to get to a billion this is about 30 and any computer scientist out there who's thought about you know just how much is a kilobyte or a megabyte or a gigabyte they'll be familiar with the idea that powers of two are nice and close to these powers of ten or more specifically powers of a thousand now what I want to do is just write all of the same things log base 10 not not approximately equal to this is actually equal to 3 log base 10 of a thousand sequel the 3 log base 10 will you tell me what's log base 10 of a million it's counting the number of zeros it ends up being about 6 and log base 10 of a billion counting the number of zeros it ends up being nine now the reason I wanted to write all of this out it just emphasizes an interesting pattern here which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 we're stepping up by steps of 10 but when we're playing the same game with 10 we're stepping up by these increments of 3 so there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 it seems like we're just multiplying by 0.3 so 10 times 0.3 is 320 we scale down by that same amount 30 we scale down by that same amount okay now this is an intuition worth remembering if you have your numbers described with one base it's basically the same as describing them with another base but there's some rescaling constant okay now the next question is going to start getting us at that direction but it's going to be framed in a way that just looks like a whole pile of algebra and again I will encourage you to plug in numbers if you want to to gain a little intuition for it so as our third to last question this will be a long lecture we have which of the following is true even just a whole pile of various possible ways to combine log base C of B times log base C of a does that equal log base B log base B of A and rather than me reading them out to you I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and it's worth thinking through a little bit [Music] we have an outstanding question from the audience which is does the bar length of the pole use some kind of log function and healthily it looks like Ben eater has gone ahead and directly answered in the form of the code involved where the chart max is math dot is that raising two to the power of a ceiling of a log base two of the maximum attempt count which I think is to say unraveling if you're looking at the maximum number I'm not I'm not great at Vanna whiting this thing if you look at the maximum number in our poll it's asking what's the log base two of that so as it crosses different powers of two then that rescales it and yes yes is the answer what a fantastically apropos question thank you Karen all right so answers are still rolling in and I think like I said I just want to give you some more time to think this too because it looks like a big pile of algebra plug in some numbers to see what seems to work well and see which answer fits [Music] [Music] [Music] okay so even if you are still thinking about it I'm gonna go ahead and grade it here and then start talking about why it's true and then also why we should care why this is an operation that actually tells us something so the correct answer which it looks like around 1700 of you got congratulations is log base C of B times log base B of A is equal to log base C of a great now that's just a big ol pile of things why would that be true how do we think about it now phase one like I said me we might plug in an example but let's try to actually think about why the example holds so let me pull out we've got this as one more of the log rules this is just repeating the end look what the correct answer turned out to be where we've got an expression for hug base C of B log base B of A and this ends up being log base C of a it's gotten rid of the bees which is kind of interesting so some examples you might plug in here would be things like let's use a different color that's huge green instead of C I'm gonna go ahead and plug in ten log base 10 of 100 which is kind of asking how many times does 10 go into a not 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of let's plug in another power of 10 it'll be nice if it's also a power of 100 so I'll do a million so this one is asking ten to a hundred to the what equals a million how many times do I multiply a hundred by itself to get to a million how many times does a hundred go into a million phrasing the same thing ten different ways now the claim is that this is the same thing as taking log base 10 of a million that if I ask how many times does 10 go into 100 and how many times does 100 go into a million multiplying those should give me the answer to how many times 10 goes into a million now just checking the numbers this certainly works 10 goes into a hundred two times 100 goes into a million three times in a multiplicative sense in that 100 cubed is equal to a million and indeed how many times does 10 go into a million well 6 now we can think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger but it's actually just saying entirely the same thing so here we're if we have a base of C in a base of B and we're trying to relate those to each other the whole statement is equal to saying that um suppose the B to the X is equal to a I've got some number B you raise it to some number X and a equals a suppose it's also the case that C to the y equals B those two together are the same as saying C to the X y equals a now that's kind of a mouthful to say out loud but if you plug in some numbers to translate what all of that is really saying in the context of the example we just did saying if you can write a hundred is 10 squared and if you can write a million as 100 cubed well that lets you write a million in terms of a power of 10 yeah so sort of asking this question how many times does one number going to another but letting you layer it on top of each other now if we rearrange that expression we get what is probably the second most important of all of our log rules the most important is this top one that when you multiply the inputs you add the outputs but the second most important which is known as the change of base formula lets us write that if you want the log base B of some value a then for what did we see you want it doesn't actually matter what log you have in your pocket if you use that other log and you take the log C of a divided by the log C of B that gives you log base B of a ok so just as an example here what this would look like is let's say I wanted to be able to compute log base 100 of things I just really want to it's not a button on my calculator but I would love to be able to do it and I want to do it in with an input like a million well even if I don't have the log base 100 button on my calculate what I can do is say I'll use the log base 10 button and evaluate what's on the inside here which at least positionally it's kind of above the 100 it has a higher altitude as we write it so this can line up with the notation a little bit that it sits on the numerator and on the bottom I use the log base 10 button that's in my calculator on the base on the 100 and then I can evaluate both of those and will give me the answer in this case it gets you 6 divided by 2 which will be 3 and if we really just think through what this is saying I know I've said it many different times but it's a it's a convoluted enough way to write things but an intuitive enough fact that I think just coming at it from a bunch of different angles can be important because like I said this is probably the second most important log rule we're asking how many times does 100 go into a million in a multiplicative sense how many times we multiply by itself but division is asking that same question in an additive sense if I say how many times does the log of 100 go into the log of a million that's what division means it's saying how many times do I add this bottom number to itself to get to the top but anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses so both of the left-hand side and the right-hand side are just saying how many times does 100 go into a million but going about that in different ways so this is extremely nice because it actually lets us compute things next time we're going to talk all about the natural logarithm which is log base E often written Ln and turns out this is much easier to compute there's nice math behind it such that if you want to come up with an algorithm that your calculator can use it's actually a lot easier to think of log base E of numbers so anytime that you need to go to some kind of calculator I don't know let's say we popped over to something like desmos always happy to have as a friend and you wanted to compute log base 10 of some number you know let's say we're doing a log base 10 of 57 and it looks like that's you know it should make sense is between 1 and 2 because 57 was between 10 and 1 hundred what's going on under the hood how is it actually figuring this out it's gonna be use somewhere in there a change of base formula which is going to be that the natural log of 57 divided by the natural log of 10 is the same thing these are two different ways of writing it so if you know one logarithm you know all of the logarithms yeah and well let's just use that fact to answer one more of our quiz questions and this will be the second to last quiz question so thank you all for sticking it through I think we will be I think you'll be pleased by the last question because the last question will actually be like a fun problem solving puzzling thing and it'll involve a lot of what we've used up to this point kind of a culminating thing so before that though just to make sure that we've got the instincts of change of base down what do we have use the approximation that log base two of ten is around ten thirds so using that approximation which of the following is approximately true log base two of X is about ten thirds of log base 10 of X log base two of X is log base ten of ten thirds of X log base two of X is log base 10 of X to the power 10 thirds or log base two of X is ten thirds times the log base 10 of X and finally none of the above so we'll give you a moment to think about that you might want to think to the chart that we were drawing earlier and thinking about log base two and log base ten as we're looking at powers of 1000 and how each of those grows that can leverage some of the intuition but I'll let you think about it stop talking you while answers are rolling in it looks like a number of people have been asking on Twitter about basically how complex numbers play into this so you know we've got general asking what if the base is imaginary we've got Kalkan asking what about complex bases since e to the X is walking around the spiral wouldn't it hit every complex number well it won't hit every complex number but your instinct that it's hitting multiple things is pretty spot-on does it make sense to talk about Logs with imaginary numbers by Mitya so to all of these questions it's actually a very nuanced question this the short answer is yes complex logarithms absolutely exist but each one of them has multiple different outputs so a good a good analogy here is how if we have the square root function if I ask something like the square root of five you know we have the convention that you always do the positive amount but that doesn't quite feel honest it feels like the right answer is to specify that there's two different outputs for the square root function two solutions to x squared equals 25 and this is actually true in complex numbers as well one of the things we talked about in a complex number lecture was that you can take the square root of I and you actually get root 2 over 2 plus root 2 over 2 I but that there's two solutions you can do plus or minus this value and so you could say that the square root function isn't a function but it's a multivalued function it always has two different outputs now something funky happens when we have exponents at play so if you're just like someone who hasn't seen this stuff don't worry we talked about it in previous lectures I'm obviously jumping around on the complexity level a lot here where if in lecture 5 I'm talking about like Euler's formula with complex numbers and compound interest and like how that plays into physics and then lecture 6 we're back to the basics of logarithms I acknowledge that might be a little bit jarring to potential audience members but just continuing on with the answer if you note that e to the 2 pi times you know some number times I this basically walks you around a circle so that the output will walk around a circle and just keep repeating as n goes from 0 up to 1 it'll walk around one cycle and end up back where you started as n goes from 1 to 2 you'll end up back where you started so for example e to the 0 sits here e to the I pi is at negative 1 but e to the 2 pi is also at 1 same with e excuse me same with e to the 4 pi i that also equals 1 so in general if you wanted to ask something like what is the log base e of 1 you know the one hand we want to say the log of log base anything of 1 should be 0 because anything to the power 0 equals that 1 but if we're letting complex numbers into the mix you would have to honestly say well 2 pi is another pretty good answer to this question because e to the 2 pi also equals 1 and same with 4 pi and you could even go in the negative direction and in general n times pi sometimes 2 times I kind of wrote that in a weird order for any integer in feels like a valid answer to this question so there's a couple of ways that you can deal with that in math and before we get back to our usual lesson just on logarithm rules this is interesting enough I kind of want to pull it up let's see what if we have complex logarithm wonderful let's zoom out a little bit great so there's this notion of what's called a Riemann surface that's basically trying to capture the idea that you have a function with multiple outputs and intuitively maybe you can understand what it's getting at where the input would be something on the XY plane and the output there's just many different output sitting there so when you have a complex logarithm you have to account for that but it's used it's actually a very useful idea to do but it takes a lot more nuance than than you might expect so with all of that hopefully that helped at least partially answer some of people's questions seem same by the way with rhythms with based negative numbers because if you're asking like negative one to the X and really noodling on what that means it's it gets you into the realm of complex numbers so you have to deal with the same multiple output idea now answers are rolling in more slowly so this seems like a fine time to grade things and the answer turns out to be you rescale it if you want to go from log base two of something to log base 10 of something it involves rescaling and one way you could think about this is with the change of base formula so let's say we have let's get rid of our key steps let's say you have log base 2 of X but we want to write it in terms of log base 10 we can write that as log base 10 of X divided by log base 10 of 2 saying how many times 2 to go into X in a multiplicative sense is the same as saying how many times does the log of 2 go into the log of X in an additive sense yeah and well what is 1 divided by the log base 10 of 2 so we're going to keep our log base 10 of X out here and from what we found earlier we found that log base 10 of 2 was approximately 3 tenths was approximately 0.3 so when we divide by it that should give us 10 thirds okay and this lines up with what we were looking at a little bit earlier with powers of 1,000 let's see where was it where was it great so when we're converting from log base 10 of something up to log base 2 you know here we were thinking of multiplying the top by 0.3 to get to the bottom but you could also think of multiplying by 10 thirds to get to the top anytime you have log base 10 of some number you just rescale it and you have log base 2 of that number and the rescaling constant comes from the log base 2 of 10 or vice versa log base 10 of 2 so that's change of base like I said it's very important it´ll X you put everything into a nice universal language and that should be all of the hint that you need for the last question which is a challenge question on this one so if anyone's been watching and they've been like I know logarithms I've got this completely down let me pull up the last question which came up on let's see it's not the AMC but it's whatever the predecessor to the AMC was I believe so it's you know it's gonna involve a little cleverness and manipulation we've got this long sequence of fractions okay you take one divided by the log base two of a hundred factorial and remember hundred factorial is 100 times 99 times 98 on the way all the way down to one so one divided by the log base two of that plus one divided by the log base three of 100 factorial plus one divided by the log base four of 100 factorial on and on and on up until 1 divided by the log base 100 of a hundred factorial so this looks rather intimidating right certainly adding fractions is never fun adding a hundred fractions seems even worse dealing with a bunch of logs of different bases seems like a pain and the factorial is playing into this so I'm just gonna give you you know given that this is the wind down time I'm going to give you two or three minutes to start thinking about this and if you don't get it it's totally fine we're gonna walk through what the answer is but I'm just gonna let people think about this final challenge question before we call it a day you [Music] [Music] you [Music] you you [Music] so can I give you a little bit more time on this one because it's definitely it's definitely fun to work out and I think if you know how to start then great but if you don't know how to start just letting yourself kind of work with a couple of the different rules that we've worked out before the change of base comes into play if you'd like to use that and just kind of keep manipulating and if it feels like you're getting something that's a little too messy see if you come at it from a different angle and because answers are rolling in a little bit more slowly now what I'm going to do is just start to describe the explanation here and then come back to grade it in just a moment here so the expression that we have looks like I just started writing it out while you guys were working on it one divided by log base two of 100 factorial plus one divided by log base three of 100 factorial and before you even start the fact the thing on the inside of the log involves a big product of stuff should actually feel good because logs like to take in things that look like products because of the most important property that we have which is the idea that it turns products into addition okay so right away you know that you're probably going to use that second thing you notice is that it's uncomfortable to have all of these different bases log base two of something log base three of something log base a hundred of something so translating it all into the common language should be helpful how this change of base work well asking how many times two to go into a hundred factorial is the same as asking how many times does the log of two divide into the log of 100 factorial it doesn't even matter what logarithm you use this log could be base ten base e base two doesn't matter this change of base formula still holds now what that means for our expression is instead of taking one divided by log base two of a hundred factorial I could write that as log of 2 divided by so let me draw a little dividing line between what I'm doing here log of 2 divided by log of a hundred factorial okay and then similarly log of 3 divided by of 100 factorial so all I'm doing here is taking the reciprocal so instead of taking log of 100 factorial over log of 2 I've inverted it log of 2 over 100 factorial because that's what this reciprocal is doing so with that as the beginning I'm going to go ahead and just grade this lock in the answers and see how see how everyone's doing so it looks like awesome around 1796 we always we always draw a little bit north of Ramanujan's number around 1,800 of you answered that it's 1 which is correct congratulations 70 of you answered 100 which we can maybe see where where that our error would have come from those of you that answered 0 that's interesting it would imply that somehow you have cancellation at play because these are all positive numbers so thinking one of them was negative so you could probably gut-check against the idea that it would be 0 41 of you want me to come up with a number fun numerical fact about 69 but I won't 28th of you said 2 and I think yeah I think that's I think the difference between 1 and 100 maybe would be interesting to try to analyze but if we go back to our answer I realize it may be a little bit confused rewriting our expression up on the top here as we add all of these things and we can continue up until log of 100 all divided by log of 100 factorial the key is that now all of the denominators are the same so we can add the numerators and the more plus sign is just going to equal something that top is going to look like log of 2 plus log of 3 plus on and on up to log of 100 all divided by the log of 100 factorial and it seems like the only little contention among those answering the question is whether this should simplify to be 1 or should it simplify it to be a hundred and really the way we can think about this is to just break down that bottom part in terms of what a factorial means I'll go ahead and do this on a bottom part since I didn't manage my paper real estate very effectively here the log of 100 times 99 times 98 on and on times 2 times 1 is the same as adding all of them and you can probably see this mostly cancels out with the top the only question you might have though is that in the factorial can we keep multiplying down until we get that one which doesn't really make a difference but before you think about it too much you might wonder hang on you know on the bottom I'm multiplying everything and I'm adding this log of 1 but on the top from what we found before I never saw that log of 1 does that make any difference and the answer is no because log of 1 is saying 10 to the power of what equals 1 and the answer is 0 so in fact taking the log of 100 factorial is the same as adding the logs of all the numbers from 2 up to 100 and it simplifies down to 1 so to those of you who got it congratulations to those of you who feel like you maybe have better intuitions for change of base formula for the fact that logarithms turn multiplication into addition that's my hope in the next time with this as a foundation of logarithms that can be pointed back to what I would love to talk about is what's known as the natural logarithm log base E and try to give an instinct for why that's something that we care about ok why is it that mathematicians just seem to really love the number E sitting in the base of their logarithms what does that buy for them why does it show up in nature so that will happen on Friday at the same time as this lecture and I look forward to seeing everyone there [Music] [Music] [Music] you [Music]
Looking forward to this. I didn't understand what logarithms were until I was in my fifties.
Can you imagine how hard it is to understand what a slide rule is for when you don't understand logarithms?
I'm still working on the concept of the natural logarithm. There must be something natural about the constant e, but it's an elusive thing.