Why are log scales so common?

Question

I am currently reading a very nice book on scales in physics. There is a discussion on the different physical scales which are based on the effect of the corresponding phenomena, the given examples are the Beaufort scale, the Richter scale , star luminosity scale, the decibel scale and the diatonic musical scale. All these scales have the common feature that they relate logarithmically to the intensity of the corresponding effect,

$$ S = A \, \ln(I) + B \, .$$

$S$ is typically a number, $A$ and $B$ are system-specific quantities that are use to set the scale of one chosen object. For example, if the keys of a piano are labeled from 1 to 12 (starting at middle C), the frequency $F$, and the numbers $N$, are related by

$$ N = \frac{12}{\ln(2)} \ln(F) - \frac{12}{\ln(2)} \ln(440) + 8 \, . $$

Here the fact that A is 440 Hz and the doubling of the frequency every 12 keys constrain $A$ and $B$.

My question is the following: Why are such log scales so common in nature? They seem to emerge in extremely diverse situations. Is there a common mechanism for this to happen independently of the particular physics involved?

Note that the Mohs scale of rock hardness seems to be a counter-example of this.

Answer 1

Exponentials come as solutions of this differential equation:

dx/dt=c*x

where c is a constant and x and t variables. The solution is of the form:

x(t)=e^(c*t)

A great number of measurements and observations we make can be approximated by this equation.

Once the solution is exponential it is logical to take the log since the numbers become large fast and will not fit optionally in a plot.

So the question becomes why this simple approximation, which leads to the differential equation is so prevalent in nature

delta(x)/delta(t)~c*x

particularly for time measurements.

Note that this is a first order equation. Suppose that the real solutions obey complicated differential equations. The first order solution , will be this lowest form for many cases of physical observations, particularly where there is no symmetry, as with time measurements. Thus it acquires the prominence you describe.

It is a bit similar to the role of the harmonic oscillator for potentials, because it is the lowest order in an expansion for a symmetric potential. That is why the harmonic oscillator has such a prominent role in approximations.

Answer 2

At the sensory/neuropsychological level, humans (and probably other creatures) detect and respond on a compressed level versus the actual intensity detected by the eyes, ears, nerve endings, etc.

Perceived brightness and loudness changes don't correspond linearly to changes in physical intensity (power/area). Also, sound pitch perception depends on consecutive multiples (a doubling of frequency produces a matching sounding musical pitch).

Examples of these perceptions: 1) Hipparchus divided stars into magnitudes, the 1st magnitude being the brightest, 2nd magnitude the next, on down through 6th. When we eventually got around to cardinalizing his ordinal system, (relating luminosity to his magnitudes), we ended up with a logarithmic compression.

2) Testing of human hearing perception on with large populations indicates that loudness follows a logarithmic behavior compared to physical intensity.

3) If you ask male and female casual singers to match a pitch spontaneously, they will most often sing an octave apart, indicating they hear an octave as being the same note even if the frequency is doubled. The octave is split by multipliers, traditionally into 12 steps (that's an totally different story).

4) The cochlea is the organ which translates vibrations into nerve signals. Current studies indicate that frequency placement along the (unfolded) length of the cochlea is logarithmic with frequency.

I would say that human biophysics of sensory receptors pushes us to explain relationships in a logarithmic fashion.

There are also specific things in nature which are non-linear and must be mathematically described as either logarithmic or exponential. Two that I can think of right now are 1) radioactive decay populations 2) voltages in cylindrical geometries.

Answer 3

Your question appears to confuse the map for the territory. The scales you mentioned don't occur in nature; they occur in humans' descriptions of nature.

Phenomena - actual earthquakes, radiation from stars, etc. in their full complexity are what occur in nature. We often make measurements based on these phenomena and get results such as the total energy released by the earthquake or the total energy received from the star. Finally, we often convert the results of the measurements using a log scale. This says very little about nature or the unity of the phenomena. It is simply a choice of how to communicate and think about the results of the measurements.

We are free to make log scales for anything. We could report our heights on a log scale, if we wanted. In practice, we usually do this when we expect the quantities to vary over many orders of magnitude. We simply report the order of magnitude directly; doing so is making a log scale.