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

This page assumes you have seen nothing. We name and picture every ingredient the parent note Exponential growth and decay models silently uses, in an order where each one leans only on the ones before it.

0. The very first symbol: a quantity that changes

Plain words. We track how much of a thing there is: number of bacteria, milligrams of drug, count of radioactive atoms. We give this amount a name so we can talk about it.

The picture. Imagine a jar filling with (or emptying of) marbles. At each instant the jar holds some number — that number is $N$ .

Why the topic needs it. Everything else describes how this one number moves. Without a name for the amount, there is nothing to grow or decay.

1. Time $t$ and the idea of "rate"

Plain words. $t$ is the clock reading — seconds, hours, years. As $t$ ticks forward, $N$ changes. The rate is the answer to "how fast is $N$ moving right now?"

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

The picture (figure above). Plot $t$ across the bottom, $N$ up the side. The curve is the story of the amount. At any single instant, the rate is the steepness of the curve there — drawn as the slope of the straight line that just kisses the curve (the tangent, orange). Steep line = changing fast. Flat line = barely changing.

Why this tool and not another? We could ask "how much did $N$ change over a whole hour?" (an average), but exponential behaviour is about what happens at each instant — the feedback is continuous, not once-an-hour. The derivative is the only tool that captures change right now, at a single moment. That is exactly why the parent starts with $\frac{d N}{d t}$ and not a table of hourly jumps.

2. "Proportional to" and the symbol $\propto$

Plain words. Two quantities are proportional when one is always a fixed multiple of the other: double one, you double the other. The symbol $\propto$ means "is proportional to".

The picture (figure above). On the left, few marbles → a short arrow of change. On the right, twice as many marbles → an arrow exactly twice as long. The length of the change-arrow tracks the pile size in lock-step. That is what $\frac{d N}{d t} \propto N$ says in a picture.

Why the topic needs it. "More stuff → proportionally faster change" is the entire physical assumption. Writing it as $= k N$ turns a sentence into an equation we can solve.

3. Why the answer is a curve: the self-feeding loop

Plain words. Because more stuff makes faster change, and faster change makes even more stuff, the pile feeds itself. This loop is why the graph bends instead of going straight.

The picture (figure above). Compare two curves. The straight line (gray) adds the same amount each step — no feedback. The exponential (blue) multiplies by the same factor each step — the gap between steps widens because each new step starts from a bigger base. Notice the constant-difference versus constant-ratio contrast: it is the fingerprint that distinguishes exponential from linear.

4. The number $e$ : the natural multiplier

Plain words. When change happens continuously (every instant feeds the next, not once per hour), the natural multiplying number that shows up is $e \approx 2.71828$ . It is the base that makes the growth rate exactly equal to the current size.

Why this tool and not $2^{t}$ or $1 0^{t}$ ? Any base would give some exponential curve, but only base $e$ makes the derivative come out clean with no extra clutter-factor. Since our law says the rate is $k$ times the amount, we want the base whose rate matches its height — that base is $e$ . See Natural logarithm and e for the full story.

5. The logarithm $ln$ : the "undo" button for $e$

Plain words. To find how long something takes (solving for $t$ when $t$ is stuck up in the exponent), we need to pull $t$ back down. The tool that undoes $e^{(\dots)}$ is the natural logarithm, written $ln$ .

The picture (figure above). $e^{x}$ and $ln x$ are mirror images across the line $y = x$ (dashed). Whatever $e$ does — turn an exponent into a size — $ln$ reverses: it turns a size back into the exponent that produced it.

Why the topic needs it. Half-life and doubling time ask "for what $t$ does $N$ hit a target?" — $t$ lives inside the exponent, so we must apply $ln$ to free it. That single step $2 = e^{k T_{d}} \Rightarrow ln 2 = k T_{d}$ is impossible without knowing what $ln$ does.

6. Putting the symbols together

Now every piece of $N (t) = N_{0} e^{k t}$ has a meaning and a picture:

Symbol	Plain meaning	Picture
$N (t)$	amount at time $t$	height of the curve
$t$	clock reading	horizontal axis
$\frac{d N}{d t}$	rate of change now	slope of the tangent
$\propto, k$	"proportional to", strength	change-arrow scales with pile
$N_{0}$	starting amount	height at $t = 0$
$e$	continuous multiplier	self-matching-slope curve
$ln$	undoes $e$	mirror across $y = x$

The machinery that combines these — separating variables and integrating — is Solving first-order separable ODEs. Its discrete cousin (multiplying by a factor each period) is Compound interest, and the classic real-world example is Radioactive decay (Physics).

Prerequisite map

Equipment checklist

What does

N (t)

mean and what does it look like on a graph?

The amount of stuff at time

t

— the height of the curve at horizontal position

t

What does

\frac{d N}{d t}

measure, geometrically?

The instantaneous rate of change — the slope of the tangent to the curve at that point.

Why must we use the derivative rather than an average change?

Because the feedback is continuous (each instant feeds the next), so we need the rate at each moment, not over an interval.

What does

A \propto B

mean, and how do we turn it into an equation?

A

is a fixed multiple of

B

; write

A = k B

with

k

a constant.

What law defines the whole topic, in symbols?

\frac{d N}{d t} = k N

— rate proportional to current amount.

What does the sign of

k

decide?

k > 0

gives growth (rising curve),

k < 0

gives decay (falling curve).

How do you tell exponential data from linear data?

Linear = equal difference per step; exponential = equal ratio per step.

Roughly what is

e

, and what special property earns it its place?

e \approx 2.718

; the function

e^{t}

equals its own rate of change, matching the law

\frac{d N}{d t} = k N

What question does

ln y

answer, and what does it undo?

"To what power must I raise

e

to get

y

?" — it undoes

e^{(\dots)}

, so

ln (e^{x}) = x

Why do half-life/doubling-time problems need

ln

The unknown time sits in the exponent;

ln

is the only tool that pulls it back down so we can solve for it.

What are

ln 1

and

ln \frac{1}{2}

ln 1 = 0

and

ln \frac{1}{2} = - ln 2