Editing E (mathematical constant) (section)

=== Exponential growth and decay ===
{{Further|Exponential growth|Exponential decay}}
[[Exponential growth]] is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous [[Rate (mathematics)#Of change|rate of change]] (that is, the [[derivative]]) of a quantity with respect to time is [[proportionality (mathematics)|proportional]] to the quantity itself.<ref name=":0">{{cite book|chapter-url=https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions |chapter=6.1 Exponential Functions |title=College Algebra 2e |publisher=OpenStax |first1=Jay |last1=Abramson |display-authors=etal |year=2023 |isbn=978-1-951693-41-1}}</ref> Described as a function, a quantity undergoing exponential growth is an [[Exponentiation#Power functions|exponential function]] of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as [[quadratic growth]]). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing [[exponential decay]] instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different [[exponentiation|base]], for which the number {{mvar|e}} is a common and convenient choice:
<math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.</math>
Here, <math>x_0</math> denotes the initial value of the quantity {{mvar|x}}, {{mvar|k}} is the growth constant, and <math>\tau</math> is the time it takes the quantity to grow by a factor of {{mvar|e}}.