Logarithms & Exponential Growth

The unifying principle

Compounding is the statement that the rate of change of a quantity is proportional to the quantity itself:

$\frac{dN}{dt} = rN$

Integrating gives the continuous form:

$N(t) = N_0 \, e^{rt}$

and the discrete (per-step) form, when growth happens $n$ times with factor $(1+r/n)$:

$N(t) = N_0\left(1 + \frac{r}{n}\right)^{nt} \xrightarrow{n\to\infty} N_0 e^{rt}$

The logarithm is the exact inverse: it recovers the number of compounding steps (time, doublings, digits) from the result.

$t = \frac{1}{r}\ln\!\frac{N(t)}{N_0}$

Two invariants recur everywhere:

• Doubling time: $t_2 = \frac{\ln 2}{r}$ — the time to multiply by 2, independent of starting size.
• Log-linearity: plotting $\ln N$ against $t$ gives a straight line of slope $r$. Exponential curves become lines under logs; that's why we can see compounding.

How it shows up in each field

Biology — bacterial growth

In exponential phase, each cell divides on a fixed schedule, so population size obeys $N(t)=N_0\,2^{t/g}$ where $g$ is the generation (doubling) time. Equivalently $N(t)=N_0 e^{rt}$ with $r=\frac{\ln 2}{g}$.

Why it's the same idea: division rate is proportional to the number of cells present → $dN/dt = rN$.

Stock-Market — compound interest

An investment reinvesting its returns grows by a fixed fraction each period:

$A = P\left(1+\frac{r}{n}\right)^{nt} \to P e^{rt}$

Why it's the same idea: interest is earned on the whole current balance, so $dA/dt = rA$. Log returns $\ln(A_t/A_{t-1})$ are used precisely because they add across periods where percentage returns multiply.

Coding/CS — Moore's Law & algorithmic complexity

Two faces of the same math:

Moore's Law: transistor count doubles ~every 2 years: $T(t)=T_0\,2^{t/2}$ — pure exponential growth.

Algorithmic complexity: logarithms measure how many times you can halve a problem.

Why it's the same idea: divide-and-conquer compounds a factor of $\tfrac12$ per step; $\log$ counts the steps.

Why this bridge matters

• Doubling-time intuition transfers everywhere. A biologist's feel for "how many generations?" is identical to an investor's "Rule of 72" and a programmer's $\log_2 n$ depth. Learn it once, use it four times.
• Log-scale is the universal linearizer. Bacterial growth curves, stock charts, and Moore's-Law plots are all drawn on log axes because compounding becomes a readable straight line — slope = growth rate.
• Small rates, huge outcomes. Intuition from finance (a 1% edge, compounded, dominates long-run) explains why a slightly faster-dividing microbe outcompetes rivals and why an $O(\log n)$ algorithm crushes $O(n)$ at scale.
• The inverse is the insight. In every field the exponential is the phenomenon and the logarithm is the analytical tool: recovering time, doublings, or steps from the outcome.

Connections

• 01 Exponential & Logarithmic Functions
• 02 Differential Equations — Growth Models
• 03 Bacterial Growth Phases
• 04 Compound Interest & Log Returns
• 05 Moore's Law & Scaling
• 06 Big-O & Divide-and-Conquer
• 07 Rule of 72 & Doubling Time

#bridge