Exponential growth and decay models

Real-world models of the form N = N0 e^{kt} (or A = a + b e^{kt}): finding the starting value, fitting the rate constant to data with logs, interpreting parameters, rates and limiting values.

Pure · Exponentials & logarithms

Exponential growth and decay models

Real-world models of the form N = N0 e^{kt} (or A = a + b e^{kt}): finding the starting value, fitting the rate constant to data with logs, interpreting parameters, rates and limiting values.

Why it works

Anything whose rate of change is proportional to its current size — populations, radioactive mass, a cooling drink, savings with continuous interest — follows an exponential model, usually written

N = N_0\,e^{kt}.

Reading the parts is most of the work:

$N_0$ is the value at $t = 0$ (substitute $t = 0$ ; since $e^0 = 1$ , $N = N_0$ ).

It is the starting amount.

$k$ is the rate constant. $k > 0$ is growth, $k < 0$ is decay. To find it you

need one more data point: substitute, isolate the exponential, then take logs.

Many models are shifted, like $A = a + b\,e^{kt}$ or $A = a - b\,e^{kt}$ . As

t \to \infty

with

k < 0

the exponential dies away, so

A \to a

: the constant

a

is the long-term limiting value the model can never pass. That ceiling is often the reason a model is "not appropriate" once a target exceeds it.

Finding the rate from data. Given a known

(t, N)

pair, rearrange to

e^{kt} = \tfrac{N}{N_0}

, take

\ln

, and divide:

k = \tfrac{1}{t}\ln\tfrac{N}{N_0}

.

Interpreting in context is examined heavily: state what a value means (a heart rate, a temperature, a number of bacteria), include units, and remember the model's domain — extrapolating far beyond the data, or past the limiting value, is where models break down.