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Exponentials and Logs - Overview

In mathematics, the exponential function is the function ex, where "e" is the base of natural logarithms. It is a special case of the natural logarithm, which is the inverse function of the exponential function. The exponential function is defined for real arguments x by the power series: The exponential function is a periodic function with period 2. The exponential function is an entire function, which means that it is differentiable for all x and its derivative is nonzero for all x. The exponential function maps the real numbers onto the non-negative real numbers. The exponential function is a special case of the hyperbolic cosine function. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = ex is upward-sloping, and increases faster as x gets larger. Its slope is positive, since ex has a positive slope. The exponential function is also used to model exponential growth, in which a constant change in time gives the same proportional change in some other quantity. The graph of y = ex is upward-sloping, and increases faster as x gets larger. Its slope is positive, since ex has a positive slope. The exponential function is also used to model exponential growth, in which a constant change in time gives the same proportional change in some other quantity.

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Video Transcript

Welcome to our math review video on exponential and logarithms again, This is for physics 101 So we're gonna look at these functions specifically with how they apply. Tow our physics courses? Not necessarily From the mathematicians point of view. If you really don't remember these functions, I highly recommend that you go look at the numerator course that covers them. Um, that will give you a much broader sense of how to use them. I'm gonna be talking them about about them in a very specific way, and you'll see what I mean here in a moment. So when we're talking about exponential functions, the generic way of expressing an exponential function looks something like this. F of X is equal to some constant A to the power of X. Well, it turns out that in physics, generally speaking, when we use thes, we're not going to use a to the X so much as we're gonna use this constant E to the X. If you remember from your math classes, E is a numeric constant. Just like pie. It's approximately equal to 2.72 It has all kinds of useful properties and shows up in a lot of different places and including in physics. Usually when we use an exponential function, it's gonna end up half E as this constant here. Similarly, when we use log rhythms will use a base e in our log rhythmic function. So what are exponential is used for? Well, if we look at the plot, it can kind of give us an idea. Uh, exponential plots look something like this, where, as theme independent variable gets larger, the function e to the X or eating whatever the independent variable me well, we often use time and physics instead of X. But whatever it is as that independent variable gets larger, the function increases faster and faster and faster. Um, this is often used to model things like populations in biology's, whereas a population grows its ability to grow increases, and because of that, it has a rapidly increasing rate of growth that's based on the size of the population itself. Um, eso in physics, uh, that occurs occasionally, though not nearly as often is in biology. In fact, in physics more often what we're interested in isn't the function e to the X, but the function e to the negative x which will look something like this. Okay. And what this shows is that as X gets larger as the independent variable gets larger, that this value gets in smaller, faster and faster and faster. In fact, it approaches zero. And what this is used for in physics is for something called a damping coefficient, where we'll have a function that describes they a force or something like that. And as time goes on, that force gets smaller and smaller and smaller until it completely disappears. And we will use an exponential function and e to the negative X function in order to model that sort of action in particular. Think of something like a pendulum that swings back and forth and back and forth but gets gradually theano attitude of the motion gets smaller and smaller and smaller until it completely stops moving. We can model that sort of motion with this kind of function. Um, so that's the place. It probably shows up most often in physics, though there is another, more complex way that it shows up literally complex. Um, some of you may have run into if you've been taking your intro Cal courses something called Oilers function. Okay, um, or Oilers identity or their steering? I've seen it called a couple different things. Um, so oilers function. What it means is that we have e to the I theta is equal to cosign if data plus I sign of data where I is. In fact, the imaginary number the route square root of negative one. Um, So you might ask yourself, why is it that we would use imaginary numbers to model riel things? Well, it turns out when you use ah, complex number, a complex number is a combination off a real term added into an imaginary term, or rather, a term that has three imaginary number in it. Um, when you use a complex number like this, you can actually model all sorts of riel motion and come up with correct solutions, because this could be used to solve all kinds of differential equations, which will mention briefly in a later video. Um, but because of this, we can use it and you can see that it's related to cosign data and science data. So this e t. The data is actually used for modeling all sorts of oscillate. Torrey Motion so again, thinking about the pendulum going back and forth and back and forth in this constant motion, you can use this in order to model that kind of motion in all any kind of motion that's oscillating back and forth in this kind of pattern. Um, now, when we run into Exponential is there's some basic properties that will use to solve these functions. Uh, first of all, we have e to the X, all to some constant C. This can be made equivalent to e to the C Times X. Okay, so these are identical things. This is one of the properties of exponential. Also, we have e to the X Times e to the Y can be equal to eat to the X plus. Why could be very helpful to combine two factors into one term, and then we also can have e to the X over e to the Y can be rewritten as eat the X minus. Why all three of these properties can be used effectively for solving problems where these functions show up and we'll practice doing that in the later example video. Let's move on to logarithms, where we will find that logarithms are actually what's called the inverse of exponential. Remember that logarithms are usually written as L O G. Of X, with some sort of base here. Usually we default to base 10 when you first start learning math, and so we'll have logged Based 10 of X is how we would read that in physics, though very often, As I've said before, we'll use the base of E log based E G of X, and in fact, we can write Log based off of X. It's such a common function that actually is often it is the natural log. That's what we call this. So this is the natural log rhythm function. It's equal to log based E of X Now. As I said before, logarithms are the inverse function of exponential meaning. If we have the natural log of E to the X, that's going to come out as just X or if we have e to the natural log of X, then that is just X notice that I'm matching the constant here with whatever the bases in the log rhythm, so you could do the log, based 10 of 10 to the X would be equal to X. Um, So let's consider again, because this is the inverse. What is it gonna look like? Well, it's gonna look something like this, whereas the independent variable gets larger. Is the independent variable gets larger, the dependent variable. This natural log of X function is going to get is going to decrease the rate that it goes up, and that rate of decrease is going to get harder and larger. In other words, the slope of this function gets smaller and smaller and approaches zero eso. This could be really useful for for a couple of different applications. The one you're going to see in physics 101 is when we go to calculate decibels and when we calculate decibels, there is a log function inside of that. So we should also learn the properties of logs, so we'll be able to handle that. So properties of logs looks something like this. If we have the natural log of a times B that is equal to the natural log of a plus, the natural log of be so we can see that multiplication. Rather, if we have a product of variables inside the log function, we can actually separate that out in term two terms. Or if we had two terms that both had logs, we could combine them down to one. Similarly, if we have the natural log of a over B, that would be equal to the natural log of a minus the natural log of be. So these two properties, they're going to be the ones that show up most often. One property that can show up, though, isn't quite as common. Looks something like this say we have natural log of A to the B. Then we can actually pull the be out in front of it and have be times the natural log of a all by itself. So thes three properties in particular, are going to cover 90% of the instances we run into where we need to solve equations that involve log rhythms. One more thing, I'll mention, is that there is a way in order to change bases, which can be really helpful. So, for example, if we have logged the base a B of X that could be equal to log face a of X divided by log face a of be and you can use this equation to switch between bases. If you need to do that, hopefully this was helpful and we will dive into some examples here in the coming video.