Other notable logarithmic scales

The logarithmic scale recurs in physics, chemistry and astronomy. Here are four notable examples and a useful counterexample.

Example — Richter scale of earthquakes

The Richter magnitude of an earthquake is $M = \log_{10}(A/A_0)$, where $A$ is the maximum amplitude of the seismic waves and $A_0$ a reference. An earthquake of $M=7$ has an amplitude $10$ times greater than one of $M=6$ and $100$ times that of one of $M=5$. The energy released grows as $10^{1{,}5 M}$: one degree more $\approx 32$ times more energy.

Example — pH in chemistry

The pH of a solution is $\text{pH} = -\log_{10}[\text{H}^+]$, where $[\text{H}^+]$ is the concentration of hydrogen ions in mol/L. A neutral solution ($[\text{H}^+] = 10^{-7}$) has pH $= 7$; a strong acid with $[\text{H}^+] = 10^{-1}$ has pH $= 1$; a base with $[\text{H}^+] = 10^{-13}$ has pH $= 13$. A difference of one pH unit corresponds to a factor $10$ in the concentration.

Example — Apparent magnitude of stars

The apparent magnitude $m$ in astronomy is defined by $m_1 - m_2 = -2{,}5\log_{10}(F_1/F_2)$, where $F_i$ is the light flux. A difference of $5$ magnitudes corresponds to a factor $100$ in flux (because $-2{,}5\cdot\log_{10}(1/100) = 5$). The brighter stars have a smaller magnitude (Sirius $m = -1{,}46$; Sun $m = -26{,}74$).

Remark — Is the Mach number a logarithmic scale?

No: the Mach number $\text{Ma} = v/v_{\text{suono}}$ is linear, not logarithmic. It is a useful counterexample for understanding the difference: one has a logarithmic scale $\iff$ the numerical value is the logarithm of a physical ratio.

Links

Topics: Logarithmic function
Concepts: Logarithm · Apparent magnitude · pH · Logarithmic scale · Richter scale
Functions: Logarithmic function
Skills: Interpret graphs · Model