Bacteria, earthquakes, loudness, money: real quantities can grow so violently that on ordinary graph paper the early history flattens into the axis while the recent past shoots off the top. Here you plot the genuine data and then bend the ruler: switch the y-axis (or both axes) to a logarithmic scale, where each step means "ten times as much" instead of "one more." Watch the impossible curve snap into a straight line — and read its growth rate straight off the slope. That is the whole job of a logarithm: it makes explosive change legible.
A linear axis measures differences; a log axis measures ratios. On ordinary graph paper, moving up one centimetre always adds the same amount — 10, 20, 30, 40. On a logarithmic axis, moving up one centimetre always multiplies — 1, 10, 100, 1000. Equal spacing means equal factors, not equal amounts. That single change is why a quantity spanning a millionfold range fits on one screen: each power of ten gets its own fair slice of height.
Why exponentials go straight. Exponential growth means a fixed percentage increase per step: \(y = a\,e^{rx}\). Take the logarithm of both sides and it becomes \(\ln y = \ln a + r x\) — the equation of a straight line, with the growth rate \(r\) as its slope. So plotting \(\log y\) against \(x\) (a semi-log plot) turns any exponential into a line. If your data straightens on semi-log paper, it is growing exponentially, and the tilt tells you how fast.
Why power laws need both axes. A power law, \(y = a\,x^{k}\), describes things that scale — an animal's metabolism with its mass, a city's infrastructure with its population, the area of a shape with its length. Take logs of both sides: \(\log y = \log a + k\log x\). Now both variables are logged, so only a log-log plot straightens it, and the slope is the exponent \(k\). Semi-log for exponentials, log-log for power laws — the shape that straightens tells you which law you're looking at.
The scales you already use are logs. The Richter scale, decibels, the pH scale, stellar magnitudes, and the musical octave are all logarithmic — humans invented them precisely because the underlying quantities (earthquake energy, sound intensity, acidity, brightness, pitch) vary over ranges too vast for a linear number to be useful. A magnitude-6 quake isn't twice a magnitude-3; it's about 30,000 times more energetic. The log scale is what lets a single, comfortable number stand in for that avalanche.
Reading a rate from a slope is the working scientist's payoff. Fit a straight line to a semi-log plot and its slope hands you the growth (or decay) constant directly — this is how half-lives, epidemic doubling times, and Moore's-law trends are extracted from noisy data. On a log-log plot, the slope is the scaling exponent, which is how physicists spot power laws hiding in everything from earthquakes to word frequencies.
A caution worth keeping. Because a log axis compresses, it can make wildly different growth rates look similar and can hide the sheer size of the final numbers — a gentle-looking upward line may still be a catastrophe. The log scale is a lens for seeing structure (is it exponential? what's the rate?), not for judging magnitude. The honest analyst shows both, and says which ruler they used.
What is computed for real: the plotted points are the exact model values — \(N_0e^{rt}\), \(P(1+r)^t\), \(10^{bM+c}\), \(I_0\,10^{L/10}\), \(aM^{k}\) — sampled live and mapped through whichever axis transform you select (identity or \(\log_{10}\)). The green line is a genuine least-squares fit performed in the current axis space, and the reported R² is the real coefficient of determination — that is why it jumps to ~1.000 exactly when you pick the ruler that matches the law.
Reading: any introductory data-analysis or physics-lab text covers semi-log and log-log paper and "linearizing" a fit. For power laws in nature, see Kleiber's metabolic-scaling law and West/Brown/Enquist's work on allometry; for the log scales in the world, the Richter (energy \(\propto 10^{1.5M}\)) and decibel (\(L=10\log_{10} I/I_0\)) definitions.
Sibling labs in this repo: the exponential story connects to #3 Taylor series (how \(e^x\) is built) and #4 the Lorenz butterfly (Lyapunov time is a log-slope); the "read a slope" habit echoes #23 eigenvalues and the fitting in #17 Monte Carlo.