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