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In this lesson, our instructor Vincent Selhorst-Jones will teach you how to solve exponential and logarithmic equations.--you probably wonder.0285Why can you do this--how can you get away with using Pe can morph into other forms.0297For example, let's look specifically at a possible half-life formula.0303We might have P times 1/2 to the t/5; we can see this as P times 1/2 to the rate of 1/5 times t.0307That is what we have there: some principal starting amount, times 1/2 to the 1/5 times t.0318So, for every, say, 5 years, we have half of the amount there that we originally had.0323So, how can we get Pe form right here; we know what r is, so we swap that in for our r.0343And we get P times e to the -0.1386 times time.0350But we also know that -0.1386 is the same thing as -0.6931 times 1/5.0358So, if we want, we can break this apart into a -0.6931 part and a 1/5 times t part that we might as well put outside.0366We have e to the -0.6931, to the 1/5 times t, because by our rules from exponential properties, that is the same thing0374as just having the 1/5 and the -0.6931 together, which is the r that we originally started with.0381Now, it turns out that e to morph into something else.0396We can get it to morph into this original 1/2; and now, we have this 1/5 here,0402so it becomes just P times 1/2 to the t/5, which is what we originally started with as the half-life formula.0406So, by this careful choice of r--and notice, the r here is equal to 1/5; the r here is equal to the very different 0.1386;0414we get totally different r's here; but by choosing r carefully, if we have enough information from the problem0426(sometimes you will; sometimes you won't; you will have to know that special formula)--sometimes,0433you will be able to get enough information from the formula, and you will be able to figure out what r is.0438So, you can have forgotten the special formula--you can forget the special formula occasionally, when you are lucky.0441And you would be able to just use Pe, we morph it into something that works the same as the other formula.0448Now, of course, you do have to figure out the appropriate r from the problem.0454You are just saying it--you have to be able to get what that r is.0457And remember: it is the r for Pe, which may be (and probably is) going to be totally different0460than the r for the special formula that we would use for that kind of problem.0465But if you can figure out what the r is from the problem, you can end up using Pe instead.0469Once again, we will talk about a specific use of this in Example 2, where we will show how you can actually use this if you end up forgetting the formula.0473Now, I want you to know that the above isn't precisely true.0480e isn't precisely 1/2; it is actually .500023, which is really, really, really close to 1/2; but it is not exactly 1/2.0484But it is a really close approximation, and it is normally going to do fine for most problems.0496It is such a close approximation that it will normally end up working.0500And if you need even more accuracy, you could have ended up figuring out what r is, just to more decimal places.0504And you could have used this more accurate value for r.0509Applications of logarithmic functions: logarithms have the ability to capture the information of a wide variety of inputs in a relatively small range of outputs.0513Consider the common logarithm, base 10: if we have log(x) equaling y, log(x) going to y--0522over here we have our input, which is the x, and our output, which is the value y--that is what is coming out of log(x).0528x can vary anywhere from 1 to 10 billion; and our output will only vary between 0 and 10.0537That is really, really tiny variance in our output, but massive variance in our input.0545Why is this happening?Because 1 is the same thing as 10, we will get 10; and as it operates on everything in between, we will get everything in between, as well.0577So, there is massive variance in our inputs and massive different possible inputs that we can put in.0585There is a very, very small range of outputs that we will end up getting out of it.0589This behavior makes logarithms a great way to measure quantities that can be vastly different--0594things that can have really huge variance in what you are measuring.0599But we want an easy way to compare or talk about them; we have to be able to talk about these things.0602They come up regularly, and we don't want to have to say numbers like 10 billion or 9 billion 572 million.0607We want some number that is fairly compact--that doesn't require all of this talking.0614So, we use logarithms to turn it into this much smaller, more manageable number that makes sense, and we can understand, relative to these other things.0618Earthquake magnitude is one of the things that is measured on it.0625It is measured on the Richter scale, which is a logarithmic scale.0629Sound intensity is another one; it is measured in decibels, which is another logarithmic scale.0632Acid or base concentration is measured on the p H scale; we will actually have examples about that in Example 3.0637And that one is measured, once again, on a logarithmic scale.0643And many others--there are many other logarithmic scales,0646when you have a really, really large pool of information that can be going in as an input,0649but you want to be able to narrow that to a fairly small, manageable, sensible range of values.06530 to 10 is going to have lots of decimals, when you evaluate log of 8 billion and 72 million.0658It is going to have lots of possible decimals to it, but it is going to be a fairly small, manageable number for thinking about.0664Logarithms will also show up in formulas that are analyzing exponential growth,0670because if we are building a formula that is going to be connected to exponential growth,0675if we are trying to break down and figure out what its power is raised,0678we are going to end up having logarithms show up when we are solving it;0682so they will end up showing up in the formula, as well.0684Logarithms show up in formulas for analyzing exponential growth.0686All right, let's get to some examples: A principal investment of 00 is made in an account that compounds quarterly.0690If no further money is deposited, and the account is worth 57.57 in 5 years, what will it be worth after a total of 10 years?Vincent begins with an introduction and then applications of exponential and logarithmic functions. Youll learn why logarithms are useful and then get five chances to practice with them.Lecture Slides are screen-captured images of important points in the lecture.Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.Hi--welcome back to 0000Today we are going to talk about applications of exponential and logarithmic functions.0002At this point, we have a good understanding of exponentiation and logarithms.0007In this lesson, we will see some of the many applications that they have.0010Exponential and logarithmic functions have a huge array of applications.0013They are used in science, in business, in medicine, and even more fields.0017They are used in all sorts of places.0021There are far too many applications to discuss them all in this lesson, so instead we will focus on working a variety of examples.0024We will begin with a brief overview of some other uses--some of the uses that we can see for exponential functions and logarithmic functions.0029Then, we will look at many specific examples, so we can really get our hands dirty and see how word problems in this form work.0035Now, before you watch this lesson, make sure you have an understanding0040of exponents, logarithms, and how to solve equations involving both, before watching this.0043We won't really be exploring why the actual nuts and bolts of this solving works--how these things work.0048We are just going to be launching headfirst into some pretty complicated problems.0053So, you really want to have an understanding of what is going on, because we are going to hit the ground running when we actually get to these examples.0057Previous lessons will be really, really helpful here if you are not already used to this stuff.0062OK, let's go: applications of exponential functions: exponential functions allow us to describe the growth or decay0066of a quantity whose rate of change is related to its current value.0073So, how fast it is changing is connected to what it is currently at.0078So, some examples of applications: we can also see how their rate of change is related to its current value:0083Compound interest--the amount of interest that an account earns is connected to how much money is already in the account.0088If you have ten thousand dollars in an account, it will earn more than if it has one thousand dollars in the account,0096or than if it has one hundred dollars in the account.0101So, this is an example of seeing how the rate of change is related to the current value of the object.0103Other things that we might see: depreciation--loss in value; compound interest and loss in value0109are both used a lot in banking and business--anything that is fiscally oriented.0114Population growth is used a lot in biology; half-life--the decay of radioactive isotopes--shows up a lot when we are talking about physics.0118If you are studying anything in radiation, understanding half-life is very useful.0127And many others--there is a whole bunch of stuff where exponential functions are going to show up.0131It is really, really useful stuff.0135Now, I have a secret for you: don't let anybody else know about this.0138Many exponential functions have their own formula--things like compound interest, 1 plus the number of times that it compounds in a year, divided...0142Oh, this should actually be the other way around; it should not be n/r; it should r/n.0152The rate of it, divided by the number of times it compounds, to the number of times it compounds, times the time--0160if you don't remember that one, remember our very first lesson on exponential functions that described why that is the case.0167Population doubling is P, some original starting principal amount, times 2 to the rate that they double at, times t.0173Half-life is some principal starting amount times 1/2 to the rate times time.0181However, if you forget all of these formulas--there are a bunch of different formulas;0187there are even more than just these; but it is sometimes possible to use the natural exponential growth model.0191You can sometimes swap out any one of these more difficult-to-remember ones for simple "Pert"--0197P, the original starting amount, times e, the natural base, to the r times t,0205where r is the rate of the specific thing that we are modeling, times time.0210r will change, depending on what different thing you are doing.0214So, even if you are modeling isotopes--half-life in plutonium and half-life in uranium--you will get very different r's,0217because the plutonium and uranium will have different rates of decay.0222So, you are not going to use the same rate r.0226Once again, if you are talking about half-life, the r here would be totally different than the r in our Pe form from the problem, you can get away with using it instead.0246There are many situations where you might not remember any one of these specialized formulas.0255But it can be OK if you have enough information from the problem to be able to figure out what r has to be.0260There are lots of cases where that will end up being the case.0266We will talk about a specific one on Example 2; we will see something where we could get away with not knowing0269the specific formula, and still be able to figure things out by using this Pe formula.0275We will talk about it in Example 2, if you want to see a specific example of being able to use this secret trick.0279Why is this the case?
0707Well, we have compounding, but not continuous compounding; so we go and look that up.0710It is principal, times 1 the rate, divided by the number of times that compounding occurs,0714raised to the number of times that the compounding occurs, times the amount of time elapsed in years.0721So, our principal investment here is P = 4700; and we know that, at time 5, at t = 5, we have 5457 dollars and 57 cents.0727So, 57.57 is equal to..was our principal amount? 0743That is one of the things we don't know yet--we don't know what our rate is.0754And that is why we are setting up at t = 5, instead of just hopping immediately to the 10 years question:0758we need to figure out what our rate is first, so that is what we are figuring out now.07631 r/n; our n is quarterly, so that is an n of 4, because it happens four times in a year, in each of the four quarters of the year.0767So, 1 r/4, raised to the 4 times t--do we know what t is in this case?
That, at least, is the only reason I can imagine to explain why the absurd access to the i8 was deemed tolerable.
A knack for graceful entrances and exits does gradually develop over time, but the i8’s doors still occasionally elicited a sigh – particularly on those nights when you’d traipse off a jam-packed train, into the car park and find it hemmed in by other cars. Invest in some yoga classes to prepare yourself for when you need to post yourself through a tiny aperture to get into it (see inelegant demonstration below).
From there, it would run on to a top speed of 149mph.
The 16mpg average fuel consumption wasn’t anything special but when a car looked as good as the M6, who cared?