Exponential Growth and Decay

Exponential growth and decay is a function in which you have to multiply the initial value by a certain growth factor. 

We can use the exponential growth and decay concept for modeling situations exhibiting any growth or decay in value by specific factors over a period. The best and the most common example is interest, wherein our money increases by a particular percentage periodically.

There are two types of exponential functions: exponential growth and exponential decay. These functions are in the form:

y = abx

We see exponential growth if we have a positive a and b > 1. But if a is positive and 0 < b < 1, we see exponential decay.

What is Meant by Exponential Growth and Decay?

Exponential growth and decay are the continuous growth or decay in the pattern of the value. If there is steady and exponential decay, we call it a decay model, whereas if there is a continuous rise, we call it an exponential growth model. 

Exponential growth example includes bacteria populations that increase exponentially over a span.

Exponential decay occurs when the numbers of some entity decrease exponentially. Unlike growth, where the value multiplies, decay sees repeated and rapid division. An exponential equation is involved in this, but the values are decaying. 

The value of a car is an excellent example of this kind of model. You will notice the value of the vehicle is higher when you buy it but decreases due to depreciation over time. 

Exponential Growth and Decay Formula

An exponential growth model occurs when the entity increases exponentially and rapidly over time. The formula for this model is:

f(x) = a(1 + r)x 

Where,

f(x) = exponential growth function

a = initial amount

r = growth rate

x = number of time intervals

Exponential decay occurs when there is a decrease in value exponentially over some time. Its formula is:

Where,

N = quantity

λ = exponential decay constant

t = time

Applications of Exponential Growth and Decay

There are several applications of exponential growth and decay. Let us look at some of them.

Growth of Bacteria

As mentioned above, we can use the exponential growth model to observe the rapid growth of bacteria over time. 

Many infectious diseases exhibit the growth of bacteria. The exponential growth model helps track the initial, rapid development of these bacteria and viruses. 

COVID-19 is an excellent example of exponential growth, where the disease spreads rapidly from one person to another.

Ageing

The ageing process in living beings in the latter half of life has an exponential decay rate. 

After the age of about 60, a person’s ageing process is rapid, affecting their body exponentially. There is a degradation in the quality of life they live.  

Degradation of Food

Food degradation is another typical example of exponential growth and decay. 

Over time, stored food starts degrading and decaying. Once food decay starts, we can see a rapid pace in the degradation and decaying of the food item.

Feedback

This concept of feedback increases gradually, meaning customer feedback follows an exponential growth model. 

We can especially observe it during a negative feedback cycle. If customers face a bad experience with the product, it triggers a chain reaction leading to exponential growth and decay model.

Conclusion

Exponential growth is a rapid increase in some kind of entity in an exponential manner. Exponential decay is an exponential decrease in the values over a period of time.

The concept of exponential growth and decay means a rapid change in quantities. The most important applications of exponential function include the decay model and growth models. 

This model applies to various natural processes, such as ageing, population growth, compound interest, bacterial growth, radioactive decay, feedback, etc. This article discusses a few of these applications.