The Rt Hon Rees-Mogg, MP, and the logarithmic scale

A personal note: I am old enough to be taught how to use a logarithmic ruler and I am a proud owner of one!

It is always interesting – and occasionally amusing – when politicians use mathematics in their speeches. The UK new business minister, Mr Jacob Rees-Mogg, MP, has just referred to logarithms in his argument for allowing back fracking:

It is safe, it is shown to be safe, the scare stories have been disproved time and time again. The hysteria about seismic activity fails to understand that the Richter scale is a logarithmic scale. It seems to think that it is a straight arithmetic scale, which of course it is not.

https://www.theguardian.com/politics/live/2022/sep/22/liz-truss-therese-coffey-gps-nhs-health-uk-politics-live?CMP=share_btn_tw&page=with%3Ablock-632c39d68f089a550b8aa58a#block-632c39d68f089a550b8aa58a

I am a great fan of logarithmic scales; having even given a popular lecture on that topic some years back. But they need to be used carefully.

The Richter scale is used to describe the severity of earthquakes. Developed in 1930s, it is designed to cope with a huge range of the power released in the process. From small tremors which are hardly noticeable, to devastating catastrophic events at a global scale, we need a way to assign the numbers.

https://www.geologyin.com/2015/01/using-richter-scale-to-measure.html

To cope with such a range of energy release and hence damage, the Richter scale uses logarithms – more precisely, base-10 logarithms. Thus, the increase in the magnitude by one unit is associated with an energy increase by a factor of 10.

Logarithms were invented by John Napier, Scottish mathematician, physicist and astronomer who lived between 1550 – 1617, exactly for the purpose of dealing with both small and huge numbers. The idea is to turn multiplication (which can be quite complicated for large numbers) into addition (which is simple), and division (which is even more complicated) into subtraction (which is simple).

Thus, it turns a scale in which the value of consecutive elements increases arithmetically (by a unit) into a scale in which the values are multiplied, for example by 10. The Rt Hon Member of the Parliament for North East Somerset seems to know that and was trying to use the Napier (a Scot!) idea to explain away other MPs concerns regarding fracking.

Fracking is known to result in (relatively small) earthquakes. Basically, water and chemicals are pumped into the ground under high pressure to extract oil. The fluid fractures the rock structure and the resulting collapse can cause waves to propagate to the ground.

To prevent high damage, there is usually a threshold of earthquake magnitude above which fracking has to stop in the location. In England, fracking companies have to halt operations for 18 hours if there is a seismic event of 0.5M. The UK government has just announced that this threshold will be increased to 2.5.

I suppose Mr Rees-Mogg was referring to logarithms to point out that 2.5 on the Richter scale is actually not a very big number. Still, it corresponds to an explosion of a large bomb, 5.6 tons of TNT. It will probably not be as damaging – fracking rarely occurs near houses – but still is something to recon with.

But I am not entirely sure what he meant by comparing it to an arithmetic scale. It is absolutely true that a logarithmic scale is not an arithmetic scale – it is actually worse. An increase of 2 units on the Richter scale means the 100-fold increase in the amplitude of shaking.

What about the energy? As explained on the Earthquake Calculator page, the energy released, E, depends on the magnitude, M, but the equation

    \[E=10^{1.5M+4.8}.\]

Thus, a 0.5 earthquake releases 354 kJ energy, and a 1.5 earthquake releases 11,220 kJ – an increase of about 32 times. For a 2.5 earthquake, the increase is a staggering 1,000 times (to 354,813 kJ)!

Now, fracking might – or might not – be the way forward in the time of high oil and gas prices. It also might – or might not – be resulting in problems by causing earthquakes. But, it is still very important to understand the mathematics behind the politician statement, particularly if mathematics is being abused.

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