History
John Napier is credited with discovering/inventing logarithms but nature had already beaten him to it.听 Our bodies had already figured it out. Our senses鈥攕ight, hearing, taste and touch鈥� use a logarithmic transform to cope with the enormous range of the signals our senses need to handle.
Bargains & Biology
Take shopping. $10 off a $20 pack of coffee is a fantastic bargain! Saving $10 off a $200 dress? Same $10 saved but this one barely registers. We react instinctively to proportional change; the difference divided by the amount. The same applies to our senses.
Our eyes can adapt to dim starlight to bright sunshine鈥� an intensity range of approximately 100 trillion times, or 1014. Our ears can pick up the tiniest whisper, up to roaring jet engines, a sound range of approximately 10 trillion times, 1013. Over a more modest range the other senses, touch and taste behave the same way. Peppers have their .
So how on earth do our bodies deal with this huge range? They adopt the 鈥渂argain鈥� approach, the ratio, the logarithmic compression. Sound intensity in听听is something you will be familiar with. Biology scientists talk of the . Volcanologists report earthquakes on the logarithmic . Economists use , a related concept.
Things take off exponentially. You hear this all the time, something 鈥渢aking off exponentially!鈥� And it usually does. Exponential growth is sneaky. Numbers start to increase slowly but then they start to rocket at an ever increasing speed. Logarithmic scales transform a set of exponentially increasing numbers into a corresponding set of simple linear ones. These are much easier to follow but, as we shall see, there are serious issues that this can create.
1听 10听 100听 1000听 10,000听 100,000听 1,000,000听 10,000,000听 100,000,000
100听 101听听102听 103听 104听 105听 106听 107听 108听鈫�
The top line gives the exponential increasing numbers written out. The bottom line shows these as exponents of ten. The exponent numbers are the logarithms, 1, 2, 3, 4鈥�.14
To see how this works, we鈥檒l choose the one example where we really feel it. Have you ever wondered why you never hear of Richter one to four? These report micro-earthquakes that are much too small for us to notice These are only detected by seismometers. At the other end, we quickly notice when a quake reaches the higher numbers 5 to 8/9, exponentially, shaking, like hell.
Logarithms and Risk Perception
How do we assess risk/hazard? Here鈥檚 the leap. It slowly dawned on me that we also estimate听risk听using the same logarithmic shortcut. I鈥檝e worked early on as a consumer advocate for CHOICE and Canberra Consumers and then later presided over government activities, such as removing asbestos in the mid 1980s to later as chair of a Commonwealth government agency [now the AP&VMA pesticides & veterinary medicines & much else] in the mid 90s.
From this experience, I suspected that humans also use a logarithmic scale to get a quick gut feeling about risk. If true, this follows the pattern seen with all the examples above. Conclusion; we dramatically overestimate low, and dramatically underestimate high risks.
Logarithms sacrifice accuracy for convenience
A change in the set of small numbers, greatly and falsely, appears to be much more important than the change in the large numbers. Both ends of the logarithmic scale are highly distorted.
Large risks. Two classic examples
We are warned not to look directly at the sun during solar eclipses. Why? While the sun鈥檚 light intensity is greatly reduced, percentage wise [around 72% - 98%], this doesn鈥檛 seem a big deal on the logarithmic scale that our senses use. We ignore this large, but apparently small change, at our peril.
We are also warned to avoid or protect ourselves against loud noise. Why? Hearing also responds logarithmically to sound intensity. When the volume goes belting up in the blaring music at the disco, logarithmic suppression dulls our response to what we think is only a small increase. The exponentially increased damage to our hearing can chronically become permanent. So far, this has been about the distortion at the large end of the exponential to logarithmic transition. What about the small end?
Small risks, many examples
We can worry excessively about pesticide residues in food [trivial], or bits of solid asbestos found in a park [clean up without panic], more than about chronic diseases like diabetes, heart disease, or obesity [huge]! Richter makes obvious the reality of the small end of the logarithmic scale鈥� by its absence! Repeating. The earth rumbles are too weak for us to notice in Richter 1-4, and so you never see entries for them. In contrast to earthquakes, for assessing risk we lack an inbuilt 鈥渟haking鈥� that would make us aware of our distorted perception.
So while transitioning from poacher (a consumer advocate) to a gamekeeper (involved in assessing how to respond to public agitation on issues) for example as once the Chair of the Board of the Australian Commonwealth regulator of pesticides and veterinary medicines, swimming chemical pools, and much else, I came to the realisation that the distortion at both ends distorts the resources allocated to deal with risk by regulation. Over time, this has dampened my youthful career as a 鈥減oacher鈥�, and swung it more towards trying to be fair, sensible and equitable 鈥済amekeeper鈥�.
So that鈥檚 it for now folks. Decompress your logarithms when thoughtful contemplation is needed for safety鈥檚 sake.
Ben Selinger is an Emeritus professor of chemistry at the Australian National University, Canberra.
