Semi-log plots

A semi-log plot graphs data with the vertical axis on a logarithmic scale (equivalently, it plots $\log(y)$ against plain x). On that axis, the gridlines 1, 10, 100, 1000 sit at equal spacing, because each is one factor of 10 above the last. Moving up a fixed distance anywhere on the axis multiplies the value by a fixed factor. The distance from 10 to 100 equals the distance from 100 to 1000: both are one factor of 10, even though one covers 90 units and the other 900.

That is the whole trick, and the whole hazard. Vertical position now encodes factors, so additive habits, "that point is 50 higher," stop meaning anything. On a semi-log plot, higher by a fixed distance means larger by a fixed multiple.

Take logs of an exponential model $y = a \cdot b^x$ and the product rule unrolls it:

1. $\log(y) = \log(a \cdot b^x) = \log(a) + x\log(b)$.
2. That is a line in the variables x and log(y): slope $\log(b)$, intercept $\log(a)$.

So exponential data, and only exponential data, plots as a straight line on semi-log axes. The line's two numbers recover the model's two parameters: read the intercept, undo the log to get a; read the slope, undo the log to get b. A slope of 0.301 with intercept 0.699 (base-10 logs) means $b = 10^{0.301} \approx 2$ and $a = 10^{0.699} \approx 5$: the data behind the line is $y = 5 \cdot 2^x$. Decay works identically: $y = 80 \cdot (0.5)^x$ gives a straight line with negative slope $\log(0.5) \approx -0.301$.

The plot is a diagnosis tool precisely because it does not straighten everything. Linear data like $y = 5x + 3$ bends on semi-log axes: early on, each step of x multiplies y by a large factor (3 to 8 is a factor of 2.7), but later the same +5 barely nudges the multiple (503 to 508 is a factor of 1.01), so the curve rises steeply and then flattens, concave down. Logarithmic data bends even harder. Only a constant-factor-per-step pattern earns a straight line.

So the reading discipline runs in both directions. Straight line on semi-log axes: the data is exponential in its original units. Curve on semi-log axes: it is not. The classic misread is the first one inverted, "the plot is a line, so the data is linear." The line lives in log coordinates; back in original units it is the signature of multiplication.

Real datasets spanning huge ranges, populations from 3 to 3,000,000, are unreadable on ordinary axes: the small values flatten into the floor. The log axis gives every factor of 10 the same vertical room, which is why epidemiologists, audiologists, and seismologists live on semi-log paper. It reveals multiplicative structure; it does not manufacture it.

Two quick reads you should own. Recover the model from two points: a semi-log line through (0, 4) and (3, 32) shows a factor of 8 across 3 steps, so the per-step factor is the cube root, $r^3 = 8$ giving r = 2, and the model is $y = 4 \cdot 2^x$. Read constant doubling time: if a bacteria count's semi-log line climbs one factor-of-2 gridline every 3 hours, the count doubles every 3 hours, at hour 30 exactly as at hour 3. Raw counts rise faster and faster, but the doubling time never changes; that steady rhythm is what the straight line is showing you.