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Logarithmic scales

Logarithmic scales are incremented by the base-ten exponent of 1. This results in a scale which compacts the intervals between data points progressively.

This is useful for data which is unevenly distributed, and a scale with equal interval spacings would be impractical, and would obfuscate relationships between data points.

The scale is incremented as non-linearly. If the scale is base-10, the increments and actual values are:

Scale measureIntensityValue
0$10^0$1
1$10^1$10
2$10^2$100
3$10^3$1,000
4$10^4$10,000
5$10^5$100,000
6$10^6$1,000,000
7$10^7$10,000,000
8$10^8$100,000,000
9$10^9$1,000,000,000
10$10^{10}$10,000,000,000

Well-known logarithmic scales are:

Decibels

Decibels is a logarithmic unit, expressing a physical quantity as a ratio with a base, reference quantity:
$$L_{dB} = 10log_{10}({P1}/{P_0})$$

where $L_{dB}$ is the relative intensity, in decibels, of the ratio of $P_1$, the measured power, and $P_0$ the reference power.

A decibel is one-tenth of a bel, a unit named in honour of Alexander Graham Bell, the telephone pioneer. The unit was first developed as a way of measuring signal loss in telephone lines.

The decibel unit is used for amplifier gain, signal attenuation, signal-to-noise ratios, and sound pressure. If the sound intensity is weighted to the sensitivity of the human ear for certain frequency ranges, the scale is called $dB_A$.

Sound Intensity

The logarithmic scale means that an increase in 3 dB doubles the sound intensity (perceived loudness), and the sound intensity of 80 dB is ten times the intensity of a sound of 70 dB.

$$L_I = 10log_{10}(I/{I_0})dB$$

where I is the sound intensity, $W⋅m^{-2}$, $I_0$ is the reference sound intensity $10^{-12} W⋅m^{-2}$.

Earthquake intensity

The Richter Scale is a measure of the intensity, or energy released, on a scale from 1 to the maximum recorded 9.3 Earthquake in Chile. A 5.0 earthquake has ten times as much intensity as a 4.0 earthquake, and one-tenth the intensity of a 6.0.

$$M_L=log_{10}A−log_{10}A_0(δ)=log_{10}[A/{A_0(δ)}]$$

where $M_L$ is a dimensionless measure of the earthquake intensity, relative to an arbitrary reference quake, $A$ is the maximum excursion (movement of the needle) of the Wood-Anderson seismograph, $A_0$ is an empirical function based on the epicentral distance, δ.

Introduced in 1935, by Charles Richter, an American seismologist, the Richter Scale underwent several modifications and versions, until it was finally replaced by the MMS, Moment Magnitude Scale, in the 1970s. The MMS is a dimensionless scale, based on the mechanical work done by the seismic moment, $M_0$ (N⋅m):

$$M_W = 2/3log_{10}(M_0) - 6$$

On this scale, the largest earthquake ever recorded, the Valdiva, or Great Chilean, Earthquake, of May 22 1960, registered 9.5. This corresponds to an energy release equivalent to 2.7 Gt of TNT (4.0 EJ). The largest nuclear bomb, Tsar Bomba, released 50 Mt of TNT equivalent, so the Chilean Earthquake contained the energy of 54 of these bombs.

Acidity pH

Acidity is a measure of hydrogen ion activity $a_{H+}$ in a solution.

pH$ = -log_{10}(a_{H+}) = log_{10}(1/{a_{H+}})$

Being the inverse of the positive ion activity, as the solution becomes more acidic, the pH decreases logarithmically. A solution of pH 4 is ten times more acidic than a solution of pH 5. A solution with pH of 7 is neutral, meaning the active hydronium ions ($OH^-$) and hydrogen ions ($H^+$) are equal, and greater than 7 means the solution is alkaline (a base).

Brightness of Stars

HR Diagram

The Hertzsprung-Russel diagram is a graphical demonstration of the relationship between surface temperature and luminosity. It maps the Main Sequence stars, showing that as the temperature increases, from right to left, the luminosity increases. In the top right-hand corner are the Red Giants. In the bottom left-hand corner are the White Dwarfs.

Hertzsprung-Russel Diagram
Hertzsprung-Russel Diagram, showing the stages of stellar evolution

Be careful not to interpret the graph as meaning that a star will 'evolve' along the line of main sequence. The position of the star on the main sequence is fixed by its size. When the main sequence stars have exhausted a critical percentage (this depends on their size) of the hydrogen fuel they are born with, if they are large enough they will become Red Giants, and then White Dwarfs.

Levels of Acquaintance

There is an interesting idea that asks the question: how many direct acquaintances are you from every other person on Earth?

If you know a certain number of people, and they all know a certain number of different people, and those people know a certain number of other different people, how many links in this chain of unique acquaintanceship does it take to have a high probability of connecting to everyone on Earth?

Let us take the population of the Earth as 7.4 billion (2015 estimate). Let the number of unique people (i.e. people who would not be counted by any other people as an acquaintance) we know be x, and the number of links in the chain be n.

When n = 2, I know x people who in turn each know x people. Hence at n=2, we have x2 people covered. Clearly xn = 7.4 billion.

Whether it is possible or not, depends on agreeing on the number of unique people everyone on Earth knows. Anthropology, based on a median size of tribes, and studies of such things as Christmas card lists, suggests we each have an acquaintanceship of about 150 people. (Claims of 500+ friends on Facebook fall into the 'yeah, sure' category for our purposes.)

From this number we need to decide how many would include members of our set in their acquaintance set. Obviously, a remote village in a snowbound mountainous region would have a higher coincidence of acquaintanceship than people in a large city. For the sake of making an initial estimate, let us take one-third as an average, so we have x = 50.

So, 50n = 7.4 x 109.

n = log507.4 x 109
= log107.4 x 109/ log1050
9.87/1.7 = 5.8

If x is smaller, say 20, the result is: 7.6

Even taking a very low number for x, say 5, n remains a modest, sub-rugby team length of 14.

Acquaintances graph
Graph of the inverse log relationship between the number of unique acquaintances each person has on average and the number of people needed to have personal links to all people on Earth.

Our guess is that probably an estimate of 20 unique people is reasonable, so within 7 or 8 people we know the prime minister of Australia, as well as Mrs Ipsofacto, a factory cleaner in lower something-on-the-whatever, Anonymoland. And probably more disturbingly, they know us...

Like all the best things about maths, this has no practical application whatsoever.

However, it is food for thought in terms of how computer viruses and viral promotion of ideas and messages might work through social media. How many links in the chain does it take Facebook and email group distribution networks to ensure the cybersphere is informed of a news item by word-of-click?

Content © Andrew Bone. All rights reserved. Created : April 3, 2015 Last updated :February 14, 2016

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