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Vincent begins with an introduction and then applications of exponential and logarithmic functions. You’ll learn why logarithms are useful and then get five chances to practice with them.Lecture Slides are screencaptured images of important points in the lecture.This rather complex formula shows you how to solve this puzzle using accepted scientific methods.
After 5600 years, if we start with a gram, we end up with half a gram. you probably wonder.0285Why can you do thishow can you get away with using Pe can morph into other forms.0297For example, let's look specifically at a possible halflife 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 halflife formula.0406So, by this careful choice of rand 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 formulayou 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 ityou 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 y0522over here we have our input, which is the x, and our output, which is the value ythat 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 different0594things 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 compactthat 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 othersthere 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? If you have a fossil, you can tell how old it is by the carbon 14 dating method. This is a formula which helps you to date a fossil by its carbon.

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