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Exponential Decay

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Today we're going to talk about exponential decay. So we'll start by looking at how we can derive the formula for exponential decay. And then we'll do some examples using that formula.

So let's compare our general form for an exponential equation with our formula for exponential decay. Here we have some base b raised to a variable exponent and being multiplied by some coefficient a. The formula for exponential decay looks very similar, the differences being that b is our rate of change expressed as a decimal. And it's being subtracted from 1 to represent decay.

So let's do an example involving exponential decay. Suppose I take 100 milligrams of a certain medication. And I know that this medication dissolves in my bloodstream at a rate of 12% per hour. So I want to write an equation using my formula to represent this situation.

So I know that my initial amount is 100 milligrams. I took 100 milligrams. And it's dissolving in my bloodstream at a rate of 12% per hour. So my rate of change is 12%, which as a decimal is 0.12. Bringing down the rest of my terms, this gives me the equation for the situation where y is the amount of drug in my system at any time and x is the amount of time. In this case, it will be in terms of hours.

So now I can use this equation to determine, for example, how much of the drug will be left to my system after 24 hours. So I want to how much of the drug is left. So this is going to stay y. I'm writing the rest of my formula. Now instead of x, I'm going to put 24, because I want to know how much is left after 24 hours.

So I'm going to start by simplifying this in the parentheses. This will give me 0.88. 0.88 to the 24th is approximately 0.047. And finally, multiplying these two numbers, I find that y is approximately equal to 4.7 milligrams. So after 24 hours, I'll have approximately 4.7 milligrams of the medication in my system.

So finally, let's look at a specific instance of exponential decay, which is called half-life. So remember the formula for exponential decay. a is our initial amount. b is our rate of decay. and x is our time.

Half-life means that at a specific time interval, the amount of substance that you have is half of the amount that you had at the beginning of the previous time interval. So our rate of decay is going to be 50%, or 0.5. So in our parentheses, this becomes 1 minus 0.5 for half-life.

And since half-life is looking at a specific time interval, our x variable is going to become the number of half-lives instead of the amount of time. We still have the initial amount in the beginning. And y is still going to represent our end amount, or the amount at the end of the time period that we're looking at.

So now that we have a formula that we can use for half-life, we can solve some problems. So let's suppose we have 1,000 milligrams of Magnesium-27. Magnesium-27 has a half-life of approximately 9.45 minutes. So we want to know how long, how many minutes it's going to take before we have 100 milligrams of that Magnesium-27.

So looking at my formula, I know that my end amount is going to be 100 milligrams. So I'll substitute 100 milligrams for my y variable. My start amount is 1,000 milligrams. And I'm trying to determine the number of half-lives that will occur before we go from 1,000 milligrams to 100 milligrams. And then I can use that to determine how long it will take to go from 1,000 milligrams to 100 million grams.

So to solve for my x variable, I'm going to start by dividing by 1,000 on both sides. So that will give me 0.1 is equal to 1 minus 0.5 to the x. Now I can simplify my parentheses. 1 minus 0.5 is just 0.5.

And this in an exponential equation. So I can solve it by using the inverse operation of a logarithm. So I'm going to take the log of both sides. So here this becomes log of 0.1, which is just negative 1.

And here, log of 0.5 to the x exponent-- using my property of logarithms, I can write that as x multiplied by log of 0.5. My x exponent becomes a multiplier in front of the logarithm. Log of 0.5 is approximately negative 0.301. So I can rewrite this as negative 1 is equal to x times negative 0.301.

And finally, I can solve for x by dividing both sides by negative 0.301. And I find that x is approximately equal to 3.3, which again, is the number of half-lives. So we know that Magnesium-27, our substance goes through 3.3 half-lives before it goes from 1,000 milligrams to 100 milligrams.

So if we want to know how many minutes that takes, we can multiply 3.3 half-lives by our half-life of 9.45 minutes for one half-life. So when we multiply these two together, we know our half-life unit will cancel and our units will just be minutes. And so we found that for our substance to go from 1,000 milligrams to 100 milligrams, it's going to take approximately 31.2 minutes.

So let's go over our key points from today. Exponential decay equations can be used to represent real world situations such as drug filtration or half-life. In the exponential decay formula, a is the initial value, b is the rate of change expressed as a decimal. It is subtracted from one to represent decay. When solving an exponential decay equation for the variable exponent, one method is to take the log of both sides of the equation at apply properties of logs.

So I hope that these key points and examples helped you understand a little bit more about exponential decay. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Exponential Decay