Exponential Decay
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
Exponential decay refers to a situation where the value of a quantity decreases over time or as the independent variable (usually x) increases, following an exponential pattern. This occurs when the base of the exponential function is a positive number between 0 and 1 (i.e., 0 < b < 1).
An exponential decay function can be written in the general form:
Where:
a is the initial value (the starting amount when x = 0),
b is the base, which is a number between 0 and 1 for decay (the rate of decrease),
x is the exponent (the input or time).
Key Characteristics of Exponential Decay:
The value of y decreases as x increases.
As x becomes larger, y gets closer and closer to zero but never actually reaches zero (it approaches zero asymptotically).
The function shows a rapid decrease at first, which slows down over time.
Example:
Consider the function y = 1000.5^x:
The initial value is 100 (when x = 0, y = 100).
The base 0.5 means that the value of y is halved each time x increases by 1.
Thus, exponential decay describes how a quantity decreases over time at a constant percentage rate.