Hi, and welcome. My name is Anthony Varela. And today we're going to talk about exponential growth. So we're going to be using a formula to represent exponential growth. And we'll use this formula in two examples.
The first example will be about the growth of a population. And the second example will be about the spread of a disease.
So our exponential equation is y equals a times b to the power of x. So we have a base number b that is being raised to a variable exponent x. All of this is being multiplied by a.
Now let's compare this to our formula for exponential growth. Now it looks very similar. But we have y equals a times the quantity of 1 plus b being raised to the power of x.
So in our formula, a is an initial value. And then we have our growth factor. This is 1 plus b. Now it includes our rate of change. That's b as expressed as a decimal. And it's being added to 1 then to represents our growth factor. And x is a unit of time.
So we're going to be using this exponential growth formula to solve a couple of problems. Our first example is with the growth of a population. So in the year 2000, the population of Berlin was about 3.39 million people. The city's population has been growing at an average rate of 0.27% each year.
So assuming the same rate of growth, what is Berlin's expected population in the year 2050? So we're going to be using our formula for exponential growth.
Now we want to know what Berlin's expected population in 2050 is? So that's y. y equals the population to start off with. So that would be in the year 2000, 3.39 million.
And now we need to multiply this by our growth factor raised to an exponent power. So our growth factor is going to be 1 plus 0.27% expressed as a decimal, so 0.0027. And this is being raised to the power of 50, because there are 50 years between 2000 and 2050.
So I'm just going to add 1 plus 0.0027 in here to have a simplified equation. So now we need to solve for y. Well, to solve for y, we're going to apply the exponent first. This is following our order of operations.
So applying the exponent first, we have y is 3.39 times 1.144328. And now we're just going to multiply these two numbers. And so y is 3.879274.
Now remember, this is in millions of people. So I can adjust this decimal number, multiplying it by a million. So the expected population of Berlin in 2050 is 3,879,274 people.
So when solving for y using our formula for exponential growth, you want to apply the exponent first and then multiply it by a.
Our last example has to do with the spread of a disease. So our situation is that during the 14th century, the Black Death, or the bubonic plague, devastated the population of Europe.
Now at its peak in 1350, the plague had infected 1.5 million people in just two years. So if the plague had continued to spread at a rate of 18.75%, how long would it have taken to infect 4 million people?
So we're using our formula for exponential growth. And now y equals 4 million people. So I'm going to say 4 for 4 million. And this equals our initial value of 1.5 million people, multiplied by our growth factor. So this would be 1 plus 18.75% expressed as a decimal. So, 1 plus 0.1875.
And now this is being raised then to the power of x. So here is our simplified equation. We need to solve for x.
The first thing that I'm going to do is divide both sides of my equation by 1.5. This is because my variable power x is attached to the 1.1875, not the 1.5 out in front.
So dividing this through, I have approximately 2.6667 on the left side of the equation. And now I just have 1.1875 raised to the power of x on the right side of the equation.
Now to undo this variable exponent, I am going to take the log of both sides. So I can say then on the left side of the equation I have the natural log of 2.6667. And on the right side of the equation, I have the natural log of 1.1875 raised to the power of x.
You can use the common log if you want, just as long as the logs that you apply have the same base. So on the left side of the equation, I'm going to approximate this as 0.9808. And on the right side of the equation, I see an exponent inside of a log function.
So using the power property of logs, I can bring the exponent out. So have x being multiplied by then approximately 0.1719. That's evaluating the natural log of 1.1875.
So now I can isolate x through division. So x then is approximately 5.71 years.
So to solve for x, what we want to do first is divide by that a value. And then we're going to apply the logarithm to both sides. And in doing so, you're going to be using this power property to logarithms that lets you take an exponent inside of a log function and bring it outside to be multiplied.
So let's review our lesson on exponential growth. Here is our formula for exponential growth, y equals a times 1 plus b raised to the power of x, where 1 plus b is our growth factor. To solve for y, you want to apply the exponent first, and then multiply it by your a value.
To solve for x, the first thing you want to do is divide by the a value and then apply the log to both sides. And you will be using this important property for logarithms to isolate x.
Thanks for watching this tutorial on exponential growth. Hope to see you next time.