RC

# 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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Welcome to our review video on the properties of exponential XYZ. In this video, we're going to look at how exponential show up in physics and then what we can do to solve the equations that they show up in We're going to start with a very simple one, the one that does not show up until the physics 102 class. This is when we're looking at a circuit that has a resistor and capacitor inside it. Don't worry if you don't know what those are. Um, that means that we when we have this type of circuit, then we have current which we right as the variable I that the current at any particular time T is equal to the original current times e to the negative t divided by sometime Constant. That will call Tao at this time. So you can see here We've come up with this e to the negative, Uh, e to a negative times are independent variable here where the independent variable is time. And so what we want may want to do is to say, Hey, at what time will it be when we have a particular current? I So let's say we have a ni final that's equal to our initial and we want to know at what point in time will we reach this I final? So the way we conduce, that is just there's some quick algebra we have i f over I zero is equal to e to the negative t over Tau. And then the simplest thing to be to do here actually is not to use any of the log properties except to remember that the natural log is going to be the inverse of the exponential function. So if I take the natural log of both sides of this, have a natural log of I f over I zero is equal to e to the negative our natural log of e to the negative t over tau. Yeah, natural log of E to the negative t over Tau. Then I can look at this and say, Oh, well, natural log will get rid of my e here, which will leave me with the natural log of what are essentially two constants on this side equal to negative t divided by tau. And we can very quickly solve for T in this case, Negative Tao times the natural log of I s over. I zero note that I f is going to be different from I zero. In this case, we would hope it would end up being giving us a negative natural log rhythm. In that way, we would have a positive time. It would be after t equals zero. Um, so we wanna watch for that whenever we solve a problem like this to make sure that we get a physical time. So this is one way that it can be done. Um, other places that it can show up, as I mentioned before is in the damped oscillations equation where we might have something that looks like position is a function of time is equal to some constant times e to the again. We're going to have this e to the negative, uh, t over some constant tau. There's a bunch of stuff inside there that I'm not gonna worry about right now. And then we'll multiply that by a a trigger The metric functions. Something like cosine of Omega T plus five again. Don't worry what all these variables mean right now, you'll find you'll come across them as you get to this section on oscillations. The important thing is that here we might say, Oh, well, uh, be really difficult to solve for T in this case. So generally you'll get a question that will be something along the lines of. Well, when will the amplitude of the oscillation be a particular size? In that case, you don't have to worry about CoSine because the amplitude of CO signs always just gonna be one. And you could say, Oh, well, when I have an amplitude A we'll call it a final in this case that will be equal to a times E to the negative t over Tau. And we can do the same trick that we just did over here by using the fact that the natural logarithms is the inverse operation of the exponential in order to solve for our little tete.

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University of North Carolina at Chapel Hill
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