The Hidden Math of Our Senses: Why Logarithms Matter

Physical quantities can differ by factors of thousands, while our senses make those contrasts feel much smaller. Logarithms help explain why.

You measure 70 dB from each of two identical noise sources. Turn them on together. Do you get 140? No, about 73 dB. The math only looks wrong if you treat decibels like ordinary numbers.

It’s easy to make the same mistake elsewhere. We add investment returns, compare the acidity of drinks, read earthquake magnitudes, and wonder which note lies halfway between two others. Sometimes an equal step means adding the same amount. Sometimes it means multiplying by the same factor.

That’s what logarithms are for: making sense of ratios across enormous ranges, when an ordinary ruler stops being useful.

The same two numbers, two different midpoints

Take 10 and 1,000. What’s halfway between them? Basic arithmetic says (10 + 1,000) / 2 = 505. You add 495 to get from 10 to 505, then another 495 to reach 1,000. That’s the arithmetic midpoint, or arithmetic mean.

But ask a different question: what if each step must multiply the number by the same factor? Then the midpoint is 100. First you multiply by 10 (10 → 100), then by 10 again (100 → 1,000). That’s the geometric midpoint: √(10 × 1,000), or the geometric mean.

Both answers are correct. Trouble starts when we forget to ask what kind of distance we’re measuring: a difference or a ratio.

Imagine one person has $100,000 in wealth and another has $1 million. What’s the next step on that ladder: $2 million or $10 million?

$10 million feels like the more natural next rung. Going from $100,000 to $1 million seems comparable to going from $1 million to $10 million. Both are tenfold increases, even though the dollar amounts are very different.

We often judge wealth and success in relative terms, not just by the raw numbers.

For expenses, distance traveled, or pounds gained, an arithmetic scale is usually easier to work with. For investment growth, compound interest, musical intervals, and sound and light intensity, a logarithmic scale is often more useful. Equal steps on it represent equal multiplication factors rather than equal additions.

Logarithms turn multiplication into addition. The base-10 logarithms of 10, 100, and 1,000 are 1, 2, and 3. These numbers sit far apart on an ordinary scale, but at equal intervals on a logarithmic one. A logarithm answers a simple question: what power must you raise the base to in order to get this number? Since 10³ = 1,000, log₁₀(1,000) = 3.

Put your intuition to the test

What feels like the midpoint between two quantities? Try three quick experiments involving time, line lengths, and the pH and decibel scales.

Both midpoints are easy enough to calculate. But which one does your perception choose?

Why time doesn’t always split in half

In our experiment, the short flash lasts 0.4 seconds and the long one lasts 3.6 seconds. The arithmetic midpoint is 2 seconds. But divide the interval into equal multiplicative steps and you get 1.2 seconds. Which feels halfway to you?

In 2024, psychologist David Sanderson found that the answer changed depending on which intermediate intervals participants had seen. To avoid this effect, each participant in the main experiments saw only one test interval. People were more likely to classify the arithmetic midpoint as long, while responses at the geometric midpoint were split roughly evenly between “short” and “long.” The data were more consistent with a midpoint based on ratios than one based on equal differences [1].

Why our senses soften enormous differences

Imagine walking out of a dark room into bright sunlight. The amount of light entering your eyes changes dramatically, yet the world doesn’t suddenly appear thousands of times brighter than usual. Your visual system adjusts to the light. Perceived brightness doesn’t follow a simple rule of “twice the light, twice as bright.”

Our perception seems to compress the gap between weak and strong signals. Light or sound intensity may increase tenfold, a hundredfold, or a thousandfold, while the sensation changes much less. In the 19th century, Gustav Fechner tried to describe this relationship using a logarithm. Later, Stanley Stevens showed that a power function fits many sensations better. The point remains: a physical signal and our experience of it don’t necessarily change in step [2].

Music offers an especially intuitive example. Going from 220 to 440 hertz takes you up one octave. Going from 440 to 880 hertz does, too. The first increase is 220 Hz; the second is 440 Hz. But the ratio is identical: frequency doubles in both cases. Musical intervals are based on frequency ratios. That’s where logarithms come in.

So 440 Hz is exactly halfway between 220 and 880 Hz in musical terms, even though the arithmetic midpoint is 550 Hz. That’s the geometric mean at work, even if you’ve never opened an algebra textbook.

70 + 70 = 73: what decibels are really measuring

Decibels make the distinction even clearer. Here the logarithm isn’t a theory about the brain. It’s built into the definition of the scale. Increase sound intensity tenfold and its intensity level rises by 10 dB. Increase it a hundredfold and the rise is 20 dB, not 100 or 200.

When two independent sources produce the same sound intensity at a given point, the intensities add to twice the original amount. The level increases by about 3 dB: two 70 dB sources together produce roughly 73 dB. How many identical vacuum cleaners would it take for the sound to seem about twice as loud as one? Around ten! That subjective increase often corresponds to roughly 10 dB, or ten times the physical intensity. NIOSH [3] gives an approximate rule of 8–10 dB.

This matters when you’re reading headphone specifications, assessing workplace noise, or comparing readings from a sound level meter. You can’t add decibel values the way you’d add ordinary numbers.

Acidity: two pH points can mean a hundredfold difference

The pH scale looks like an ordinary acidity scale. But it’s based on the negative base-10 logarithm of hydrogen-ion activity. At pH 3, that activity is 100 times as high as at pH 5. Just two pH points hide a hundredfold change [4].

In one study, cola measured around pH 2.4 and black coffee around pH 5.1. That’s a difference of roughly 500-fold in hydrogen-ion activity [5].

For comparison, an empty stomach often has a pH of about 1–2.5, while arterial blood is maintained within the narrow range of 7.35–7.45. These values depend on the conditions, but they show just how much variation fits into a small numerical range [6] [7].

But a hundredfold difference in hydrogen-ion activity doesn’t mean that one drink tastes a hundred times more sour. Even beverages with the same pH can taste different. The type and quantity of acid, along with other ingredients, matter. pH captures the scale of a chemical difference, but it doesn’t precisely predict taste or damage to tooth enamel.

A magnitude 7 earthquake isn’t just a little stronger than a magnitude 6

Earthquake reports present another logarithmic trap. A one-unit magnitude difference corresponds to roughly ten times the seismic-wave amplitude and about 32 times the energy released. A two-unit difference means nearly a thousandfold change in energy [8].

That doesn’t mean a magnitude number tells you how strongly your particular house shook. Depth, distance, local soil, and building quality all affect the consequences. The magnitude scale still compresses an enormous range of physical measurements into relatively small numbers.

How can your average return be +25% when you’ve earned nothing?

Suppose you start with $100. In the first year your investment gains 100%, leaving you with $200. In the second year it loses 50%. You’re back to $100.

The arithmetic average of those two annual returns is (+100% − 50%) / 2 = +25% per year. Sounds great. Except your account hasn’t gained a dollar.

To find the annualized return with compounding, use growth factors: 2 × 0.5 = 1. Their geometric mean is √1 = 1, or 0% per year. That figure captures what actually happened to your money over the two years.

The arithmetic average isn’t wrong. The two returns really do average +25%. It’s just not the rate at which your capital grew. The bigger the swings in annual returns, the wider the gap between these two averages can become.

This is where logarithms become useful: they turn a product of growth factors into a sum. Instead of multiplying years of growth factors together, you can add their logarithms and convert back at the end.

When logarithms can make bad news look better

On an ordinary scale, equal distances mean equal increases. On a logarithmic scale, they mean equal multiplication factors. Both are mathematically valid. But one can be used to leave readers with a very different impression of what the data show.

Official documentation for the painkiller OxyContin displayed blood concentrations of the drug on a logarithmic scale. For pharmacologists, that’s a perfectly ordinary way to visualize drug elimination. Purdue Pharma was later accused of using graphs of this kind in marketing to make the decline in concentration look less steep. A graph can be mathematically correct and still leave a misleading impression [9] [10].

In 2014, the British website May2015.com used bar charts on a logarithmic scale to display political polling. But people normally compare bars by their height above zero, and zero doesn’t exist on a logarithmic axis. The gaps between the polling numbers looked very different from those on a standard bar chart. After criticism from Full Fact [11], the website switched to a conventional axis.

A five-second check: if an axis runs 1, 10, 100, 1,000 instead of 1, 2, 3, 4, you’re looking at a logarithmic scale. A straight rising line on that graph can represent constant percentage growth, not a constant numerical increase.

Why learn logarithms when your phone does the math?

Calculators have spared us the trouble of looking things up in logarithm tables. But they don’t tell us what the results mean. Some quantities should be averaged by addition, others by multiplication. On some scales, one tick marks a small increment; on others, it hides a tenfold change. One chart makes growth look almost vertical; another turns it into a straight line.

Instead of asking “When will I ever use logarithms?”, try a different question. The next time you hear about decibels, pH, earthquake magnitude, compound interest, or an octave, ask: are we measuring how much something changed, or how many times it changed? If the answer is “how many times,” a logarithm is probably involved.

References

1. Sanderson D. J. (2024). A bias-free test of human temporal bisection: Evidence against bisection at the arithmetic mean. Cognition, 247, 105770.
2. Mertens A., Mertens U. K., Lerche V. (2021). On the difficulty to think in ratios: a methodological bias in Stevens’ magnitude estimation procedure. Attention, Perception, & Psychophysics, 83, 2347–2365.
3. NIOSH (1975). Industrial Noise Control Manual. DHHS (NIOSH) Publication No. 75-183.
4. IUPAC (2025). pH. Compendium of Chemical Terminology (Gold Book), 5th ed.
5. Reddy A., Norris D. F., Momeni S. S., Waldo B., Ruby J. D. (2016). The pH of beverages in the United States. Journal of the American Dental Association, 147(4), 255–263.
6. Schneider F. et al. (2019). In Vitro and In Vivo Test Methods for the Evaluation of Gastroretentive Dosage Forms. Pharmaceutics, 11(8), 416.
7. MedlinePlus. Arterial Blood Gas (ABG) Test. U.S. National Library of Medicine.
8. USGS. Earthquake Magnitude, Energy Release, and Shaking Intensity. U.S. Geological Survey.
9. Edwards J. (2011). How Purdue Used Misleading Charts to Hide OxyContin’s Addictive Power. CBS News.
10. City of Lorain v. Purdue Pharma L.P. et al. (2017). Court filing and complaint (Exhibit 1). U.S. District Court, Northern District of Ohio.
11. Full Fact (2014). Bad Graphs: Is this graph doing well or badly?. October 30, 2014.