**Exponential growth** is an increase in some quantity that follows the relationship

**N(t) = A e ^{(kt)}**

where *A* and *k* are positive real-valued constants.

Before diving further into the mathematics, let’s look at a graphical example of exponential growth. This plot assumes that A = 3 and k = 1.

The function’s initial value at t=0 is A=3. The variable *k* is the growth constant. The larger the value of *k*, the faster the growth will occur.

## Differential Equation

The exponential behavior explored above is the solution to the differential equation below:

**dN/dt = kN**

The differential equation states that exponential change in a population is directly proportional to its size. Initially, the small population (3 in the above graph) is growing at a relatively slow rate. However, as the population grows, the growth rate increases rapidly.

## Exponential Growth: Example Problems

Exponential growth can be found in a range of natural phenomena, from the growth of bacterial populations to the speed of computer processors.

Problem 1: A colony of bacteria doubles its population every 4 hours. If the colony originally has ten bacteria, how large will the colony be 24 hours later?

Solution: Since the colony has an original population of 10, then A=10. Knowing that the population will be 20 four hours later, we can solve for the growth constant.

- N(t) = A e
^{(kt)} - 20 = 10 e
^{(k*4 hours)} - ln(2) = (4 hours)*k
- k =
**0.173 /hours**

Then, the growth constant can be used to determine the population’s size one day later.

- N(24 hours) = 10 e
^{(0.173 /hours * 24 hours)} - N(24 hours) =
**635**

Amazingly, the original handful of bacteria will blossom into a colony of nearly a thousand in one day’s time. That’s the power of exponential growth.

Problem 2: A client deposits $100 in a savings account at the local bank. The account grows by 1% interest, compounded annually. What will the value of the account be after ten years?

Solution: The initial size of the account is $100, so A=100. The account’s value will be $101 after one year, due to the interest. Knowing this, we can calculate the growth constant.

N(t) = A e^{(kt)}

101 = 100 e^{(k*1 year)}

ln(1.01) = (1 year)*k

k = **0.00995 /years**

To find the value of the account at ten years, t=10.

N(10 years) = $100 e^{(0.00995/years * 10 years)}

N(10 years) = **$110.46**

## References

Matthews, John A. “Exponential Growth.” 2014: 387–387. Print.

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