Exponential growth and decay

What the symbols mean

$N$

the amount at time t

$N_0$

the starting amount (at t = 0)

$k$

the continuous growth rate per unit time; positive for growth, negative for decay

$t$

the time elapsed, in the same units as k

$e$

Euler's number, about 2.71828

When to use it

Use it when a rate is given as "continuous" or when a quantity grows or shrinks by the same *percentage* each moment: bacteria, populations, radioactive material, a drug in the bloodstream. If the rate is applied once per period (yearly interest, say), use the growth-factor form or the compound interest formula.

Why it works

The equation $dN/dt=kN$ (rate of change proportional to amount) has $N=N_0{e}^{kt}$ as its solution. That is the property that makes e special: $e^x$ is its own derivative.

Common mistakes

• Using a percentage as k: 4% per hour is k = 0.04, not 4.
• Mixing time units: if k is per year, t must be in years.
• Using $e^{kt}$ when the rate is per period (5% a year compounded yearly is ${1.05}^t$, i.e. k = ln 1.05 ≈ 0.04879).