section_path: Applications/Exponential growth and decay
section_name: Exponential growth and decay
paragraph_text: Exponential growth and decay
  
  Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an 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 base, for which the number e is a common and convenient choice: x (t) = x 0 ⋅ e k t = x 0 ⋅ e t / τ. {\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }.} Here, x 0 {\displaystyle x_{0}} denotes the initial value of the quantity x, k is the growth constant, and τ {\displaystyle \tau } is the time it takes the quantity to grow by a factor of e.