Logarithmic Modeling

Pre-Calculus · Axiom Academy

pH, earthquakes, loudness — one idea runs them all: each step up the scale means ten times more. The numbers nature throws at us span a trillion-fold range — so we measure them on log scales , where each unit you climb is a fixed ×10 . Drive three of them and feel that jump. Chemists write pH = −log[H⁺] . Drag the pH and watch the hydrogen-ion concentration on the ladder: every +1 pH means ×10 fewer H⁺ ions — one full rung. The Richter scale — how much bigger? Magnitude is M = log(A/A₀) , so the ground-shaking ratio is 10^(M−5) compared with a magnitude-5 quake. Drag the magnitude and watch the wave grow: each +1 makes it ×10 taller. Decibels — the “×10 every 10” twist Loudness uses β = 10·log(I/I₀) , so intensity is 10^(β/10) times the threshold of hearing. The 10· changes the step: here it takes +10 dB to multiply the intensity by ×10 . Drag and watch the ladder climb. Same trick everywhere a quantity spans many orders of magnitude: a log scale trades a trillion-fold range for a handful of tidy numbers, and each step up = ×10 . You just felt it in pH , the Richter scale , and decibels — the very same idea also runs star brightness (magnitudes) and the Krumbein scale for sediment grain size.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.