# Difference Between Exponential Growth and Exponential Decay

Exponential growth and exponential decay are two fundamental concepts in mathematics and science that describe how a quantity changes over time. Both exponential growth and decay follow a mathematical model that is characterized by a constant rate of change, but they are fundamentally different in terms of how the quantity changes over time.

## What is Exponential Growth?

Exponential growth refers to a situation where the quantity of interest increases at an exponential rate over time. This means that the rate of increase of the quantity is proportional to the current value of the quantity. In other words, the more the quantity grows, the faster it grows.

Exponential growth is often observed in natural phenomena such as population growth, bacterial growth, or compound interest. For example, a population of rabbits in a field that is growing at an exponential rate will have more new rabbits born each day, and those rabbits will go on to have their own offspring, creating a compounding effect that causes the population to grow at an ever-increasing rate.

## What is Exponential Decay?

Exponential decay refers to a situation where the quantity of interest decreases at an exponential rate over time. This means that the rate of decrease of the quantity is proportional to the current value of the quantity. In other words, the more the quantity decreases, the faster it decreases.

Exponential decay is often observed in natural phenomena such as radioactive decay, the dissipation of heat, or the decay of a substance. For example, if a sample of radioactive material is decaying at an exponential rate, it will lose half of its original mass in a fixed period of time, and then lose half of its remaining mass in the next period, and so on, until it has completely decayed.

## Differences: Exponential Growth and Exponential Decay

One of the key differences between exponential growth and exponential decay is the direction of the change in the quantity over time. Exponential growth results in an increase in the quantity, while exponential decay results in a decrease in the quantity.

Another important difference is the behavior of the rate of change of the quantity over time. In exponential growth, the rate of change of the quantity is positive and increases over time, while in exponential decay, the rate of change of the quantity is negative

The following table highlights the major differences between Exponential Growth and Exponential Decay −

Characteristics

Exponential Growth

Exponential Decay

Definition

In exponential growth, numbers increase in value over time in an exponential fashion.

In decay, numbers decrease in value over time in an exponential fashion.

Exponent

The exponent in the equation in the case of exponential growth is usually an integer, a number that is greater than 1.

The exponent in the equation for decay is a fraction that is between 0 and 1.

Graph

In the case of exponential growth, the y-values on a graph will increase as the x-values increase.

In the situation of decay, the y- values on the graph will decrease as the x-values increase.

Examples

Exponential growth rate examples include the growth rates of several types of bacteria when conditions are optimal and before the substrate is depleted.

Decay examples include the decreasing value of a car (depreciation) over time and the radioactive decay of radioactive isotopes with time.

## Conclusion

The trend that is evident in exponential growth is increasingly large numbers over time. The trend in decay is the reverse of that seen with exponential growth and instead, it is increasingly small numbers over time.

Updated on: 18-Apr-2023

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