Hi, and welcome. My name is Anthony Varela, and today, we're going to talk about continuously compounding interest. So we'll develop a formula for continuously compounding interest. We'll use that formula to solve for an account balance and to solve for time. So with compound interest, we'll get to continuously compounding interest in a minute, but if we have $1,000 that gains 2% interest annually, we can say that the account balance equals that initial principle, so the $1,000, multiplied by a growth factor.
So we would express our percent-- so in this case, 2% APR as a decimal, r, and add that to 1 to represent growth. And we would then raise that growth factor to the power of t, where t is time in years because our annual percentage rate is an annual percentage rate that would be for a year. Now, with compounding interest, that means that a portion of the APR is applied every time interest is compounded, so we need to divide r by the number of times per year interest is compounded, so we'll call that n.
But we also need to multiplayer exponent t by n because the interest is compounded n number of times during one year. Now, that's compound interest that isn't continuously compounding. What do we do, then, with continuously compounding interest? That would mean that n is infinitely large.
So let's examine the part of our formula that has our variable n in it, and how do we make sense of this expression, when n is infinitely large? Well, that tends towards e raised to the power of r. So we know that r is our annual percentage rate as a decimal. What is e? e is a mathematical constant.
It's an irrational number, so it's decimal pattern, there is no discernible pattern to it, and it's non terminating. We can approximate it with 2.718281. So if your calculator has this value, you should always use it to get the most accurate answer, but if you don't have an e button, you can use this approximation.
So our equation for continuously compounding interest is a little bit more simple. We have A equals P times e raised to the power of r times t. So let's use this formula to solve some problems. Our first example, we have $9,500 that gains 3% interest annually. This is compounded continuously. So what is the accounts balance after four years?
Well, plugging this information into our formula, our account balance A is going to equal the principle 9,500 multiplied by e, and we're going to raise e to the power of 0.03-- that's 3% as a decimal, multiplied by four years. So the first thing that we're going to do is simplify that exponent. So 0.03 times 4 is 0.12, then we're going to go ahead and raise e to that power.
So I'm using several decimal digits. I'm going to eventually round to the nearest cent, so we can deal with that after we multiply this decimal number by 9,500. So when we multiply it by the principal balance, we see that after four years, the account is now at $10,711.22. Now, our second example is using the same principle of 9,500. The same APR at 3% percent. This is still compounding continuously, but our question is different.
We want to know how long it will take for the account to double in value, so we need to solve for time. Now, our principal balance doubled would be $19,000, and that equals our principal of 9,500 multiplied by e raised to the power of 0.03 times t. Here, we don't know what t is. So to solve for t, the first thing that we're going to do is divide the entire equation by the principal, and that's because this exponent here is attached to e. It's not attached to 9,500, so we can get rid of that.
And the 2 here on the left side of the equation makes sense. It corresponds to the time it takes to double in value. So we have two on that side of the equation. Now, what we're going to do is apply the natural log to both sides, and here, it's going to be the natural log because we have an expression with base e and natural log has a base of e. So the natural log of 2 and the natural log of this expression here.
Now, we can use a property of logarithms to bring this exponent out, and we have 0.03t times the natural log of e. That equals the natural log of 2. Well, we're going to use another log property. Here, the argument to this log is at the base, and so because the base and the argument are the same, that evaluates to 1. So I don't have to write that at all. I just have 0.03t equals the natural log of 2.
So to isolate t, what we're going to do is divide the entire equation by 0.03, and we get a value of t is approximately 23.1. So about 23 years for the account to double in value. So let's review our lesson on continuously compounding interest. Our formula, or equation is A equals P times e raised to the power of r times t, and remember that e can be approximated as 2.718281.
To solve for the account balance, simplify your exponent, then apply the exponent to e and multiply by P. If you need to solve for a time, first, what you want to do is divide by P, then you're going to apply the natural log, because your base is e. And then to isolate t, you can use these log properties to bring an exponent out and the natural log base e of e will equal 1. So thanks for watching this tutorial on continuously compounding interest. Hope to see you next time.