Hearing, seeing, and choosing in logarithms – part 2

In Part One of this series of posts, I demonstrated the non-linear “natural logarithmic” relationship between the frequency of a sound and our perception of it. Likewise, our perception of the volume of sound has this same “proportion-based” relationship, and I noted that this is true for our other senses as well. These are manifestations of what is called the “Weber–Fechner law.”

In this part, we will look at this relationship and our perception of light, and how this affects even human volition (i.e., choice or free will).

After spending far more time pondering the stars in the sky than we ever will, the ancient Greeks devised the basis for the scale of “apparent magnitude,” by which we still measure the brightness of stars today. That early scale ranged from one, representing the brightest star in the sky, which is Vega in the constellation Lyra, down to six, for the faintest stars that were still visible to them.

The modern adaptation of that scale extends it on both ends, adding levels of zero and -1, and even further negative, to measure the more intense brightness of the Moon (about -13 when full) and the Sun (about -27). On the high end, larger numbers have continually been added as telescopes got better, with the Hubble Space Telescope working the scale up to a magnitude of +30 for the dimmest stars it has detected.

Even though there have been modern changes to the scale, what the ancient Greeks correctly perceived was that the brightness level of stars appears linear. But in reality, the differences between the perception levels are actually logarithmic changes in the amount of light that reaches Earth. In the current formulation of the magnitude scale, each value going down the scale is about 2.512 times brighter in actual measurement than the next dimmer magnitude.

Here is the scale of apparent magnitude versus the brightness of a celestial object relative to the star Vega, which has a value of 1. The first graph shows the relationship on a conventional linear scale, while the second shows the vertical proportional brightness scale in logarithmic form.

Apparent Magnitude Linear Scale

Apparent Magnitude Log Scale

Because we got started counting with our fingers and toes, logarithms are still hard for our brain to consciously grasp. Mathematician Jordan Goldberg has a simplified explanation that works most of the time. If you are counting in “base 10” math (as non-mathematicians usually do), then the logarithm is basically the number of zeros. The logarithm of 10 is 1, of 100 is 2, of 1000 is 3, and so on. The numbers in-between have logarithms in-between as well, but not evenly spaced, rather more like the decreasing spacing of the frets on a guitar. As the base number goes up exponentially, the logarithm follows a straight line, as in the graph immediately above. [1]

Just as our ears respond to the logarithm of frequency changes as discussed in the prior post, the different layers of cells in our eyes respond to the logarithm of actual brightness. This allows us to adapt our vision to bright sunlight as well as being able to navigate by the light of a single match in a dark cavern. This logarithmic perception of light was a critical survival adaptation for our species. We would likely not be here without it.

The biological slide rule

There is a point to all of this “logarithm talk” I have been writing about. We tend to speak of human decision-making and volition in digital “Yes-No” terms, but you won’t find any clear “Yes-No” decision switches in the brain. Our body’s neurons work instead in logarithmic continua, like thousands of little slide rules interacting with one another, a complex “difference engine” that integrates thousands of inputs every second, each with probabilistic and continuous, rather than digital (Yes/No), outputs.

My freshman engineering school class in the late 1960s was one of the last to learn to use slide rules to perform some relatively-complex mathematics, as personal digital calculators were still very expensive. The number scales on a slide rule are arrayed in a logarithmic spacing rather than linearly. The “magic” of a slide rule is that when you slide the middle movable “log” scale against the top or bottom “log” scale, you are physically “adding” the two lengths represented together. However, because of that logarithmic scale, you are actually multiplying the two values. Logarithmic math basically turns multiplication problems into addition problems, and it turns an exponential problem (say, 2 to the 10th power) into a multiplication problem.

Slide Rule

And so, if our head’s neurons are responding to the logarithm of a noise or a light source, we have the biological underpinnings of “hard-wired” brain capabilities to calculate proportions, probabilities, acceleration, the effect of gravity, and angles. This is why a lion can calculate the speed, acceleration and trajectory of a running gazelle at a speed that our fingers typing on the best calculator could never match.

Humans use this capability as well, but usually only subconsciously. This is how the great hockey player Wayne Gretzky would, as he famously said, “skate to where the puck is going to be, not where it has been.”

Human volition/choice is also more about proportion and probability rather than “Yes-No”. While our brain tries to convince us that we either saw something or did not see something, or either heard something or did not hear something, the physical reality is that our eyes and ears have a long “messy middle” between the two perceptions. Our brain is constantly making up “guesses” on the fly as to what we saw or heard based on past memories mixed with the probability and the proportion of the sensory input perceived. As neuroscientists György Buzsáki and Kenji Mizusek note:

“Distance perception, time perception and reaction time…vary logarithmically with the length of distance and the time interval, respectively. Decision making and short-term memory error accumulation also obey the Weber–Fechner law.” [2]

This is why a good sleight-of-hand magician can fool us every time. Magicians don’t believe in magic (a bit of advice that has wide applicability, by the way). Instead they understand the creativity of the human brain to construct its own “reality,” which may or may not match the physical reality, and stage magicians play to that often-false perception. Our conscious brain wants certainty, but our bodies’ sensory organs are delivering “maybes.”

My late mother had a habit that really confused restaurant servers. When they would come by asking if she wanted a refill on her coffee, she would say, “Maybe…” Her brain was speaking a great truth.


  1. Ellenberg, Jordan. How Not to Be Wrong: the Power of Mathematical Thinking. Penguin Books, 2015, p. 113.
  2. Buzsáki, György and Kenji Mizusek, The log-dynamic brain: how skewed distributions affect network operations, Nature, April 2014, p. 264.

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