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Exponential Growth

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Today we're going to talk about exponential growth. So we'll start by comparing the formulas for an exponential equation and the formula for exponential growth, and then we'll do some real world examples.

So let's start by comparing our general formulas for an exponential equation and a formula for exponential growth. So we know the formula for exponential equation is y is equal to a times b to the x, where our base, b, is raised to some variable exponent, and then have some constant coefficient, a.

The formula for exponential growth looks very similar. y is equal to a times 1 plus b to the x. And in this formula, b is our rate of change expressed as a decimal. And we're adding that to 1 because we are showing exponential growth.

So let's do an example using our exponential growth formula. Let's suppose that I have a town that has a population of 20,000 people, and I know that the town population is growing at 8.6%. So I can use my exponential growth formula to write a formula to represent the growth of the town's population.

So we know that the general form is y is equal to a times 1 plus b to the x, where y is the population, the town's population, at any given time, x. A is going to be our value of the initial population, and b is the growth rate expressed as a decimal.

So my population at any time is going to be equal to the initial population, 20,000, multiplied by 1 plus my growth rate, 8.6%, represented as a decimal, not a percentage is going to be 0.086. And that's going to be to the x power. Again, x is my time.

So now that I have a formula to represent the changing population for this town, I can use it to solve a problem such as what would the population be after 10 years. So if I want to know the population after 10 years, I'm going to substitute 10 in for my x variable.

So my equation will become y is equal to 20,000 times 1 plus 0.086 to the 10th power. Now to solve this, I'm going to start by simplify my parentheses to be 1.086, bring down the rest of my terms. Then I'll take 1.086 to the 10th power, which is going to give me approximately 2.282.

And then finally, I can multiply 20,000 by 2.282. And I find that my population in 10 years is approximately 45,640 people.

So for our last example, let's suppose we have a virus that spreading through a town. Initially, 10 people have the virus, but the virus has a 95% infection rate. We want to know how long it will take for the entire town, 100,000 people, will be infected by the virus.

So we can model this with an exponential growth formula to solve this problem. So using our formula, we know that initially we have 10 people who are infected. And we want to know how long it's going to take for the entire town, 100,000 people, to be infected.

Our growth rate is 95%, or 0.95. And we're trying to solve for the time. So I'm going to start by simplifying in my parentheses. 1 plus 0.95 is 1.95, and I'll bring down the rest of my terms. Then I'm going to divide both sides by 10 to start isolating my x variable.

So this is 10,000 equals 1.95 to the x. So now this is the exponential equation. So I can use logarithms as an inverse operation to cancel out the exponential operation.

So I'm going to take the log of both sides. So now this equation is going to become log of 10,000 equals log of 1.95 to the x. Now we can use the property of logarithms, which says that we can rewrite this with an exponent as a multiply instead of being in the exponent.

And log of 10,000 and log of 1.95 are just values that I can find using my calculator. Log of 10,000 is just 4. And log of 1.95 is approximately 0.29. So then finally I can get x by itself by dividing both sides by 0.29. And I find that x is approximately equal to 13.8.

So let's go over our key points from today. Exponential growth equations can be used to represent real world situations, such as population growth or the spread of a virus. In the exponential growth formula, a is the initial value and b is the rate of change expressed as a decimal. It is added it 1 to represent growth.

When solving an exponential growth equation for the variable exponent, one method is to take the log of both sides of the equation and apply properties of logs. So I hope that these key points and examples helps you understand a little bit more about exponential growth. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Exponential Growth