# Natural Exponential Functions - Compounded Continuously

## How to Solve for the Power with Exponential Functions - Video

### Continuously Compounded Interest Formula

Continuously Compounded Interest Formula:

Here we have the continuously compounded interest formula. The way that I remember it is A = Pert. You can always use a mnemonic if you want.

P is the principal or the starting amount. r is the annual interest rate. We have to be careful with r, because of the time they give you r as a percentage without the percent sign. This means you need to convert the rate into a decimal. t is the number of years the principal is invested. A is the amount after t years.

### Example 1

Example 1:

Since we have a compounded continuously questions, let's down the formula. A = P * er * t. They gave us P = 1000, r = 6 and 3/4, and t = 5.

Now let's substitute each number into the formula.

A = P * er * t

A = 1000 * e0.0675 * 5

A = 1000 * e0.3375

A = \$1401.44

Substituted P, r, and t. Converted r to a decimal.

Multiplied r and t.

Simplified and rounded to the nearest dollar.

### Example 2

Example 2:

Since we have a compounded continuously questions, let's down the formula. A = P * er * t. They gave us A = 20000, r = 8, and t = 12.

Now let's substitute each number into the formula.

A = P * er * t

200000 = P * e0.08 * 12

20000 = P * e0.96

20000/e0.96 = P

20000/2.61170 = P

P = \$7657.86

Substituted A, r, and t. Converted r to a decimal.

Multiplied r and t.

Divided both sides by e0.96.

Converted e0.96 to a decimal.

Simplified and rounded to the nearest dollar.

### Example 3

Example 3:

Since we have a compounded continuously questions, let's down the formula. A = P * er * t. They gave us A = 12000, P = 1250, and t = 4.

Now let's substitute each number into the formula.

A = P * er * t

12000 = 1250 * er * 4

12000/1250 = e4r

9.6 = e4r

Substituted A, P, and t.

Divided each side by 1250

Simplified.

Now if you know natural logarithms then you could use some algebra to solve for r, but you can also use a calculator to solve this.

We can set the left side, 9.6, as equation 1, and we can set the right side, e4r, as equation 2. The point of intersection is where the two equations meet and is our answer. The point of intersection is (0.56544, 9.6).

0.56544 is the rate or 56.544%.

Here we have the steps for solving for r using natural logarithms. Remember the natural log is the inverse of e. If we take the natural log of each side the e cancels, then we divide each side by 4 and simplify. And our result is 0.56544, which is the same as the graph. So we know we did it correctly.