Exponential growth and decay models — half-life, doubling time

WHY does exponential behaviour appear everywhere?

WHAT we're modelling: a quantity $N$ (population, money, radioactive atoms, drug in blood) whose growth/decay speed depends on the current amount.

WHY this leads to exponentials: if you have twice as many bacteria, twice as many split each second. If you have twice as many radioactive atoms, twice as many decay each second. So:

$\frac{dN}{dt} \propto N \quad\Longrightarrow\quad \frac{dN}{dt} = kN$

The constant $k$ is the rate constant. If $k>0$ → growth; if $k<0$ → decay.

HOW to derive $N(t)=N_0e^{kt}$ from scratch

We start from only the defining property $\dfrac{dN}{dt}=kN$ and derive everything.

Step 1 — Separate variables. $\frac{1}{N}\,dN = k\,dt$ Why this step? We put all $N$ on one side, all $t$ on the other, so we can integrate each independently.

Step 2 — Integrate both sides. $\int \frac{1}{N}\,dN = \int k\,dt \;\Longrightarrow\; \ln|N| = kt + C$ Why this step? $\int \frac1N dN=\ln|N|$ is a standard antiderivative; the constant $C$ carries the initial info.

Step 3 — Exponentiate to free $N$. $|N| = e^{kt+C}=e^{C}e^{kt}$ Why this step? $e^{x}$ undoes $\ln$. Since $N>0$ physically, drop the modulus and write $A=e^{C}$: $N = A e^{kt}$

Step 4 — Pin down $A$ using $t=0$. $N(0)=Ae^{0}=A \;\Longrightarrow\; A=N_0$ $\boxed{N(t)=N_0e^{kt}}$ Why this step? The constant of integration must encode the starting amount — that's what makes the solution specific, not general.

Deriving doubling time and half-life

These are just special "how long until the amount changes by a fixed factor?" questions.

Key insight (WHY they're constant): the doubling/half-life does not depend on $N_0$ or on what time you start. Every fixed ratio takes the same time — that's the signature of an exponential.

Rewriting the model using half-life/doubling time

Sometimes we're given the half-life, not $k$. Substitute $k=\dfrac{\ln2}{T_d}$:

$N(t)=N_0\,e^{(\ln 2)\,t/T_d}=N_0\,2^{\,t/T_d}\quad(\text{growth})$ $N(t)=N_0\,2^{-t/T_{1/2}}=N_0\left(\tfrac12\right)^{t/T_{1/2}}\quad(\text{decay})$

Why this form is nice: after one half-life $t=T_{1/2}$, you get $N_0\cdot\frac12$; after two, $N_0\cdot\frac14$; instant intuition, no logs needed for round numbers.

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

• Natural logarithm and e — why $e$ is the natural base for continuous rates.
• Solving first-order separable ODEs — the machinery behind the derivation.
• Laws of logarithms — needed to solve for $t$ inside an exponent.
• Graphs of exponential functions — shape, asymptotes, transformations.
• Compound interest — discrete cousin; continuous limit gives $e^{kt}$.
• Radioactive decay (Physics) — real-world half-life application.

Concept Map

Hinglish (regional understanding)