Let’s say the population of an area increases by some per cent every year or the temperature of a place is decreasing by some degree celsius annually, these problems are identified as Exponential Growth and Exponential Decay. These two terms not only define the situation but also helps in better prediction of the future.

**Exponential Growth vs Exponential Decay**

**The main difference between Exponential Growth and Exponential Decay is that the first one increases over time and can elevate at a specific rate while the latter one denotes the decrease in value or at a rate proportional to its current value. Growth makes an elevated graph while decay makes a retarding graph.**

The exponential growth signifies a Growth in quantity with time. It draws a curve that is formed when some value increases at a specific rate. The graph becomes a curve in an increasing order which means the growth in something over a while.

The exponential decay means when a quantity or a value decreases at a rate that is proportional to its current value. The process denotes a negative curve and retardation towards the X or Y-axis. The power would be represented with a negative sign. It consists of the decay constant, concepts of half-life.

**Comparison Table Between Exponential Growth and Exponential Decay**

Parameters Of Comparison | Exponential Growth | Exponential Decay |

Definition | The exponential growth denotes an increase in quantity with time. | The exponential decay represents a decrease in quantity with time. |

Graph | The graph points far or not close to the axes since it elevates with time. | The graph will be close to axes or can even intersect or just touch it. |

Equation | If we say a value having some positive power, it denotes exponential growth. | If we represent an equation with power holding a negative sign, it’ll mean decay. |

concepts | Concepts like compound returns exist. | Concepts like Half-life exists. |

**What is Exponential Growth?**

Often we see things that take the pace with time. Like the consumption of food, buying of cars, vehicles, and many more. We notice that things are increasing day by day resulting in crowding. Also, if we see the stats of the population of countries, we often see a pattern. We notice a way that how a country is experiencing an increase.

Those things which increase with time, we say growth. And if they follow a pattern, we saw exponential growth. Exponential Growth means the acceleration in quantity over some time. It occurs when the instantaneous rate of change (delta, ∆) of a quantity with time is proportional to quantity.

Let’s understand with an example. A species of cat rises exponentially every passing year starting with 2 in the first year, 4 in the second year, 16 in the third year, and so on. Then we can conclude that in the 4th year, the quantity will be 256 or an increase of 2% every year.

In terms of finance, compound returns cause exponential growth. The compound method is one of the most powerful methods in this sector. This method elevates rapidly with time starting with a smaller investment. A firm will be able to analyse while having the exponential graphs handy and which are easy to understand. This makes it better to take decisions effectively.

**What is Exponential Decay?**

When value decreases concerning time, it comes under Exponential Decay. It follows a pattern, a formula that has a decay constant which decreases with values. If we see a formula, it would look like dN/dt = – λN.

Here N means quantity, lambda is a positive rate known as exponential decay constant and the ratio depicts the Quantity concerning time. The further solution will give terms like decay constant, or disintegration constant, or rate constant, or transformation constant.

The curve hence made after putting values in the formula will retard and move around axes. It can either remain parallel to the axes, it can touch them, or can even intersect to go in a negative direction.

A concept arises with respect to decay constant, half-life. It’s depicted by a formula that consists of decay constant. It’s a characteristic of exponential decay. It’s defined as the time required for decaying N(quantity) to fall to one half of its original value. It’s denoted by a symbol t with a subscript of 1/2. Also, concepts like decaying via two or more different process simultaneously exist. They are also known as decay modes, or decay channels, or decay routes.

**Main Differences Between Exponential Growth and Exponential Decay**

- The exponential growth signifies growth or increase in values over a period of time while decay denotes retardation in values.
- The growth graph elevates and can move far from axes but doesn’t touch while the decay graph can either be parallel and close, touch the axes, or can even intersect.
- A population increases with a specific percentage is an example of exponential growth while a decrease in temperature with every passing year by a rate is exponential decay.
- Compound returns give rise to exponential growth while there’s nothing like decay.
- If we take a specific equation whose power is taken positive, it’ll rise with an increase of values while if we take a negative value, it’ll decrease with an increase in value.

**Conclusion**

Every sector is using the concept of exponential growth and exponential decay. These two concepts are one of the most important in every industry. From these, the differential equation comes into account, concepts of half-life, etc. It’s important to revise and implement to make good analysis in future and about the future.

The purpose of exponential growth and decay remains the same irrespective of the fields using the concept. From marketing and finance sectors to agricultural and weather forecast technology. The formula provides immense convenience to the mathematician, or any person working in the growth sector to draw curves, plot several graphs and make decisions accordingly.

**References**

- https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1540-6261.2009.01518.x
- https://www.sciencedirect.com/science/article/pii/S0006349576856603

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