# Arithmetic Course/Non Linear Function/Exponential function/Exponential decay

## Exponential decay

Exponential decay function is a function that has the value decay exponentially

${\displaystyle N(t)=N_{0}e^{-\lambda t}.\,}$

Which can be proven as the root of a Differential Equation of the form

${\displaystyle {\frac {dN}{dt}}=-\lambda N.}$

## Proof

For a Differential Equation of the form

${\displaystyle {\frac {dN}{dt}}=-\lambda N.}$
${\displaystyle \int {\frac {dN}{N}}=-\lambda \int dt}$
${\displaystyle lnN=-\lambda t+c}$
${\displaystyle N=e^{(}-\lambda t+c)}$
${\displaystyle N(t)=N_{0}e^{-\lambda t}.\,}$