 Can you talk a little bit about logarithms? I've been having trouble understanding them. Yeah, for sure Dissolving girl, let's do logarithms. I'm glad I ended up here. I'm glad you like her. I Have never done an interview in my life. I've had a couple of police interrogations when you're younger Let's talk about logarithms Dissolving girl now dissolving girl, I'm gonna link you to a video that we did in the past logarithms But I'm gonna go over that right now. Well, not all of it, but a lot of it and there's a whole series of The whole playlist I'm gonna create Specifically in regards to logs the type of same type of stuff that we've done for trigonometry If you do cheeto trigonometry Introduction to the logs visualize an exponential log and the function is graphing. Here we go This is my sort of intro video to logs Okay, but I'm gonna go through it right now with you Just you just for you to have an appreciation for what it is Okay Let me sort myself up. I honestly heard someone say something like there's no failure You either win or you learn really stuck with me Yeah Pretty much as long as it doesn't take you out of the game, right? If it's gonna pop some an apple delicious Thank you. I'm a music music theorist and I always have to Interact with logs since the year here's log rhythmically does it the solving girl That's cool. I didn't know that I I associated music with a trick functions, right trigonometry Our perception of sound is non-linear. Wow, I didn't know that That's cool. That's super cool. I wonder why Is it the material that the ears made from or our processing abilities? We are more sensitive to higher pitches in real Rc Our perception of sound is non-linear. It's logarithmic. Hey, you would absolutely love studying micro tonality and Just an intonation. It's so incredible Dante, that's the Fletcher Monson curve which describes amplitude Sensation really in terms of pure frequency sensation we hear logarithmic if I go Wow because the octave is a Two-to-one ratio and we can hear the two-to-one as a distinctly different sound But the same pitch higher. Oh man You get me all excited What show me the grounds show me what it looks like. Let's talk about logs Let's talk about logs. Yeah, that's true. That's super cool So look the way it works logs is this in mathematics What we try to do musician here, I Wondered about this nice nice So in mathematics what we have we have the opposite of things that we can do, right? So in addition the opposite of addition is subtraction the opposite of multiplication is division, right the opposite of Something to a power of something to a certain degree is The radicals but the radicals are really the same thing this guy just goes if you're going to do this goes in a Denominator of the power, right? Now one of the thing we do in mathematics is when we have an equation So for example, let's say we have this We have y is equal to 2x plus One right, so let's say we have y is equal to 2x plus one and this graphs a linear function, right? This graphs this One and you go up to because the slope is two over one one two and over one This is this line, right? Now one thing We'd like to do as human beings. We'd like to take things apart, right? And take a look at well, what are they made from right factoring? The other thing we'd like to do is we'd like to mess around with things Switch up the order of things just to see what happens So mathematicians came along and said hey, okay, we know how to graph a line, but hey, what happens if we take the Reciprocal of this or the inverse of this we switch the x and the y right? What if our equation wasn't y is equal to 2x plus one? What if our equation was x is equal to 2y plus one, right? What does the graph of that look like and what does that do really? What does that do? Well, first of all, let's answer the question what that does what that does if you switch the position of the x and the y It takes any function, right any function doesn't make a difference what it is and it flips it About the line Y is equal to x Okay, this pen is dead. I'm gonna kill this one too. Let's do this in green. So you see this So it takes Y is equal to x So when you take any type of function and Switch the x and the y what you're really doing is you're taking whatever function you have and you're flipping it About the line y is equal to x you're letting the line y is equal to x act as a mirror Right. I like to think about it like this You take whatever function you have You put your fingers here along the line y equals x and you go whoop and you flip this, right? So the flip of this let's draw this in purple. I guess the flip of this is Gonna look like this That's my crappy way of drawing a line So the purple is this guy Flipped which is really this guy, right now whenever you were writing a function We're not gonna write x equals to y plus one you want to get the y by itself, right? So what you do is you get the y by itself X minus one is equal to 2y and then divide everything by two So y is equal to 1 over 2x minus a half, which is what we have here, right? This is the purple function And that's this guy because the y intercept is negative a half from here you go up one and over two one two One two, where is it? Oh up one and over two. So over here Okay Does that make sense? So that's what we're doing when you switch the x and y Keep this in mind. I'm gonna erase this So Let's take this out That was a linear function we drew right whoa That was a linear function we drew right We've got infinite number of types of functions, right or infinite number of functions There might be a limited types of functions. I don't know category wise that might be limited types of functions But there's an infinite type of functions, right math. Yes. Yes smiles back snack snack Snack that smiles back snack that smiles back. Nice. No makes sense Randy, how are you doing? So take a look at this? Let's say we have an exponential function Let's say we have the following Let's say we have a function called f of x is equal to 2 to the power of x and if you don't like f of x Let's use y. Oh my god Let's use y right So let's say we have a function called y is equal to 2 to the power of x That's a large function huge Right Actually, no, let's not do two to the power of x. Should we do two different? Yeah, let's do two to the power x Okay, let's make a table of values. Let's graph this function using a table of values. Okay Here's our x. Here's our y So let's just plug in values for x and find out what y is and we'll graph it here, right? First one we're going to do is zero. So when x is zero two to the power of zero is one So when x is zero two to the power of zero is one, right? When x is one two to the power of one is two x is one one two x is two two to the power of two is four two one two three four Four when x is three two to the power of three is eight Three that's four. We're up here. We're off the board. I did not expect this to be so satisfying What a golden lens This is a great. How are you doing now? We know what the graph looks like on this side? Let's see what it looks like on this side. Let's plug in values Negative one so two to the power of negative one is one over two Due to the power of negative one is one over two because anything to a negative power all the negative does it just flips it, right? Reminds me of crypto trading things Negative two two to the power of negative two is one over four So negative two is one over four. So it's here. So an exponential function looks like this, right? cool That's why is equal to two to the power of x Right, what do mathematicians do? What do human beings do? We like to flip things around mess around with things, right? Can we do something in four dimension or higher next four dimension? Here's a four dimension ready? This is me drawn a three-dimensional box at this moment It's live streaming this on Twitch weren't four dimensions. I Just came here to say God. I hate man so much You should love it powerful Are you talking about tensors? Tensors if you are man at some point. I'm gonna learn tensors fair That just blew my mind Take a look at this So what are mathematicians do mathematicians us we like to mess with things right flip things around That's why I'm learning now Tensors you're learning tensors One day one day in my retirement and when I'm like, I don't know say 92. I want to start learning tensors So hey, that's our function. What happens if we switch X and Y? I really need to know what the test is what happens if we go X Here, let me write that now. Let's do here X is equal to two to the power of Y. Oh Wow X is equal to two to the power of Y What's the graph of that gonna look like? Right, okay. Let's do table of values X Y So if we start plugging in numbers for X It's gonna be hard for us to solve for Y Right, is it not? It is like for example, let's assume. Let's line up the X. Let's assume X is one Right in here, we're gonna have one is equal to two to the power of Y. What's why? What's why two to the power of what is equal to one? Well, two to the power of zero is equal to one. So zero So we can do it that way plug in values for X and try to find Y But we get problems because we can't guess Most of the answers here, right? If we put an X is equal to two right to Two to the power of Y Well, two to the power of what is equal to two or one that was easy cool. What if we put in three? So two to the power. Oh, sorry three is equal to two to the power of what? Oh Way more difficult way more difficult, right? However, one thing we can do in math is we don't necessarily have to plug in numbers for X to find Y We can just plug in values for Y And solve for X, right? Let's do it that way. No one says we have to start at X This is an equation relating X and Y equation relating X and Y We just do values for Y solve for the X, right? I Google what a testers are then close the way Gonna be going back to school so a map is making me nervous Tests are very pretty. I think I'll try and learn them via Python nice That's what tensor flow is made for all right. You're making me envious. You're making me jealous One day. I'm gonna get into this right one day. I'm gonna get into this So let's plug in values for Y when Y is zero Two to the power of zero is one when Y is one Two to the power of one is two When Y is two Two to the power of two is four when Y is three Two to the power of three is eight Take a look. Do you see Zero one one two two four three eight zero one one two two four three eight Okay, let's try one more Why to the power of negative one two to the power of negative one is one over two. Oh, yeah Well, yeah, that's all you're doing so negative two is negative one over a service one over four, right? Because When you do this when you switch the X and Y's what you're doing is you're doing a Flip about the line Y is equal to X right? You're doing a flip about this line You Y is equal to X when you're doing a flip about the line Y is equal to X You're grabbing your function go So this guy's gonna go poof what you're doing in terms of tables the coordinate system You're switching the X and the Y's because that's exactly what you did you switch the X and Y So you switch the X and Y's right? You're okay there so far I think test of room might not be the right start for me too much Boiler plates. I'd rather do something like way to tour size So what you're doing is you're switching the X and Y that's what that means when you switch the X and Y in the function Right, so the graph of this guy looks like this one zero Two and one Four and two four and two Eight and three way over there a half and negative one A quarter and negative two right so the graph looks like this. Oh, sorry So we know what visually this looks like, right? Cool Now this is called exponential functions, right? What are we gonna call that these types of functions anti-exponentials? What's a good word for these? I'm gonna call them logs Because that's what logs are logs are the inverse of exponentials that's what logs are, okay, and What they do is Exactly right there just the inverse of exponentials log rhythmic functions Are you switching the X and Y around for exponential functions and coming with that function? Now what we need to do is wait. It's not Gradient descent and line is linear regression. Okay Exponential functions were the hopes of every Every technician knows crypto trader. Yeah Any trader not just crypto any trader, right? So take a look at this. This is our function, right? And again like the line that we had, right? We had the equation of line and we switched the X and Y but we don't want to write the equations as X is equal to 2Y We need to get Y by itself. How do we get Y by itself? Let's do it Just like exponentials Log rhythms have certain rules, right? So let's take this function here, let me erase this part Give us room here so we can mess around with it Let's see how dark is this that's dark that will show up Good Hard to erase store this one's nasty. Let me find one that's going to be easy to work with Okay, that's the same color. We'll use this one. So take a look at this. Let's take function X is equal to 2 to the power of Y Just like mathematics, right? You can do things on one side with an equal sign As long as you do them to the other side. So what we're going to do is we're going to take logs of both sides So this becomes log X is equal to log 2 Y Okay So all we've done right now because you need a little bit of preliminary Log intro to this but all we've done right now is take logs of both sides It's like saying here let's multiply both sides by 5. This would be 5y is Equal to 5 times 2 to the power of X, right? We just multiply both sides by 5 Okay What we're doing right now is we're taking logs of both sides Now logarithms just like exponentials. They have certain rules, right? One of the rules in logarithms is this the standards says this if you have log of a to the power of B You can kick the B down and this would be B is equal to log A Oh, sorry not equal to is equal to B log A. Okay, that's one of the rules we have Regarding logarithms. So for this right now what we can do is Grab the Y And kick it down in front of the log. So right now you got log X is equal to Y log 2 right And the name of the game is we want to get Y by itself. So we're going to divide this side by log 2 And we're going to divide this side by log 2 So this becomes a Y is equal to log X over oops Log 2 can you see that? Yeah, you can see it Now one of the things we have with this is The rules of logs is this So you can write the exponent in front of the log exactly If it's in the power it can come down to the front of the log, right? And the and the trick with learning logarithms is Learning the log rules, which is basically Learning the same way you did with how to deal with exponentials, right? So if you had exponentials like X squared times X cubed well, you add those guys that's X to the power of five once you know that You know it it's over, right? Logs has the same type of rules associated with it Learn the log rules Everything else is easy It becomes ridiculously easy For example here one of the log rules we have is this if we have log Of a over log of b Yeah, let me put the small case b you can write this as log a to the base b It's just terminology, right? Well cool this one means i'm just going to erase this part You can write this as log X to the base two So this function written in log form and I'd like to show it this way, right? If you want to convert this to a log form You can just grab the two kick it down in the log base and this just becomes Let me write it with this Y and this guy drops Y is equal to Log base two of X This is this graph okay This part you can think about it this way grab the two kick it down in the base and this guy drops Oh Focus Focus there we go Right grab the two kick it down into the base And this guy drops Okay That's the basics of logarithms Aside from this there's like Seven log rules that we have you just have to know how to manipulate them, right? How to work with them just like you did exponentials just like you do this Okay Really I know logs takes a lot of people out of the math game. It shouldn't it shouldn't It's Once you wrap your head around what logs are Then you're just playing around with another type of function And there's certain rules associated with logs and you can manipulate your function accordingly. Okay