Modeling with logarithmic functions

An exponential function turns equal input steps into equal output factors. A logarithmic function is its mirror image: equal input factors produce equal output steps. If multiplying x by 10 always adds the same amount to the output, whether x went from 1 to 10 or from 100 to 1000, you are looking at a log.

Check a table for it the same way you check ratios for an exponential, just with the roles swapped. Inputs 1, 10, 100, 1000 with outputs 5, 8, 11, 14: each tenfold jump in x adds exactly 3 to y. That is the signature. Equal input steps adding equal amounts would be linear; equal input steps multiplying the output would be exponential. Equal input factors adding equal amounts is logarithmic, and nothing else does that.

The workhorse form is $f(x) = a + b\log_{10}(x)$. Each parameter has one job:

1. a is the value at x = 1, because $\log_{10}(1) = 0$ wipes out the second term. Not the value at x = 0: the model is not even defined there.
2. b is the amount added per factor of 10 in x. Multiply x by 10 and the log rises by exactly 1, so the output rises by exactly b. It is a per-factor step, never a per-unit slope and never a percent growth rate.

To fit the form to data, anchor a at the x = 1 reading, then count factor-of-10 jumps: if f rises from 5 at x = 1 to 11 at x = 100, that is two factors of 10, so $2b = 6$ and b = 3. The single most common error in this topic is reading b as "the amount f grows per unit of x." A log model has no such number: its per-unit growth shrinks continuously, which is exactly why the per-factor step is the parameter worth naming.

Several famous measurement scales are logarithms wearing uniforms: pH, decibels, and Richter magnitude all report the log of an underlying quantity. On every one of them, a step on the scale is a factor underneath. One pH unit is a factor of 10 in hydrogen ion concentration, so pH 4 water carries not twice but $10^2 = 100$ times the concentration of pH 6 water (and lower pH means more acidic). Ten decibels is a factor of 10 in sound intensity. One Richter unit is a factor of 10 in ground motion amplitude.

The trap is reading scale steps additively, as if pH 6 minus pH 4 meant "a little more of the same stuff." Whenever a scale is logarithmic, translate differences on the scale into factors underneath before saying anything quantitative. The natural log, $\ln(x)$, changes none of this: it is simply the log with base $e \approx 2.718$, so a model $g(t) = 2 + 1.5\ln(t)$ adds 1.5 per factor of e in t, and behaves like every other log model.

A model with two free parameters can be forced through any one data point, and through any two. So "my log model passes through (10, 5)" validates nothing: it is a statement about the model's flexibility, not the data's shape. Real validation checks the whole pattern: do equal input factors add equal output amounts across the table? Do the residuals scatter without a trend when the model is subtracted off?

Shape vibes fail for the same reason. Plenty of families rise and slow down: logs, square roots, saturating exponentials. "It slows down, so it is a log" picks a family by silhouette. The discriminating question is always the signature test, because only the log adds equal amounts per equal factor. And one more habit worth keeping: a log model has no ceiling. $f(x) = 6 + 2\log_{10}(x)$ passes 8 at x = 10, 10 at x = 100, 12 at x = 1000, and keeps climbing forever, ever more slowly. Slowing is not stopping.