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Change is like a weather – it needs an adjective. Let’s first focus on a progressive change: growth.
In Oregon’s Malheur National Forest lies an exceptional example of growth. An organism so massive that its size can only be truly seen from the sky: fungus Armillaria ostoyae is quietly claiming hillsides.
In late 80’ a forest service employee named Greg Whipple noticed that pines in the forest were dying in great numbers. Initially, he made his observation from the ground, but it was only after taking a literal helicopter ride that he could appreciate the full scale of the damage: clusters of dying trees, stretched across areas spanning tens of square kilometers. The pathogen which caused the plaque appeared to be a mushroom, Armillaria ostoyae. Although mushrooms can live for many years, for most of their lifespan they remain hidden underground. Only when the right combination of humidity and temperature occurs, they produce fruiting bodies above the soil. It was at that right time that scientists could begin collecting the samples across the forest. Soon after mushroom picking came unexpected result of the genetic analysis: all pieces of collected fungal DNA were matching. It wasn’t many organism of the same species - it was a single colossal organism. Given this mushroom grows at the rate of ~1m/year, its estimated age is between 2000-8000 years. If the upper range was closer to the truth, the core part of the mushroom would be on par with the oldest known proto-cities. It would dwarf age of Stonehenge, pyramids and terracotta army. Growth and endurance at the same, isn’t it the most desired combination?
Extent of Armillaria ostoyae in Oregon’s Malheur National Forest. Genets represent different samples. Source: Big Think
Growth is a very hot word on its own. Everyone wants growth: personal, economical, political. As we usually observe growth in time – school year, quarter, lifetime, generations, it is natural to look at it as a function of time, growth(t). In today’s post I would like to have a look at few basic mathematical functions and think in which cases, if anytime, they can be used to approximate type of growth, or what else they can describe.
All plots below are interactive, you can see how the value changes by adjusting time range with the slider. As an extra you get inset with area under curve, AUC, expressed in %.
1. Constant Function
First assumption: nothing changes. If you move slider along the x-axis representing time, you see that the readout on y-axis is always the same. A real life example: financial prognoses for the new year. They are typically released in December by a number of financial institutions. No matter from which institution the analysis originates, the forecasts usually differ only in degree, not in direction. One might think it is a consensus of experts. Well, not entirely. Instead, it is nihil novi, a case of recency bias. The analysts conservatively assume that nothing will change within a year, even though a direction changing event might happen during a single press conference 1. We are still in the beginning of 2026, so it is fair to call the prognoses casted in the end of 2025: gold will go up, stocks not so much (although no one roots for bear market2), bonds definitely not as they don’t expect the interest rates to drop, or not by much. Such scenario is not much different from the situation in the 4th quarter of 2025, when these forecasts were likely written.
2. Linear Function with Positive Coefficient
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If you cycle at 10km/h, you will cover the distance of 10km in 1h. If your speed is 20 km/h, you will only need 30min to cover 10 km. Similarly, level of mercury in tuna is linearly dependent on tuna’s age. The older the fish, the more heavy metal it carries. Finally, if you deposit 1000 euro at a savings account with 1.2% yearly interest, you will get 12 euro after a year. Hold it for 2 years and you will be richer for the total of 24 euro (assuming the interest from 1st year do not capitalize). Actually, it is difficult to find truly linear outside-physics examples. The linear dependency on time is simple, elegant and therefore so rare to encounter.
On the contrary, we often assume linearity of outcome in effort-related tasks, a relation that rarely is true. If we had one more person to do this work, with two we would do it twice faster; if I work 2x longer on this post it will be 2x better. If I run 3x a week, I will see steady improvement in my body build in the weeks to come. It is rarely the case, because it assumes that time is the only factor. As if onboarding costs, fatigue, or metabolic adaptation did not exist.
3. Logarithmic Function
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Fatigue and metabolic adaptation will cause post quality and body build improvement to plateau. As such, function that better describes personal effort outcomes is a logarithmic function. An example of personal effort is skill acquisition. Make it learning a language or playing an instrument. When you start, you see progress quite quickly. And Thanks God, because learning is hard enough to have the reward delayed. If you start learning Chinese, after three months you will be significantly better in that language than the great majority of people in the street, assuming they never learned Chinese. As the time progresses, the growth in the unit of time diminishes and the function plateaus. Proficiency has limits – Shakespeare wasn’t 10x better in English than a literate native speaker. However, native speakers are very easily 10x better than adepts who only start learning the Shakespearian language. In other words, when reaching a decent level of proficiency, it is very hard to improve in a meaningful way. For example, it would cost me inhumane effort to have all the articles right in this post. Still, the progress would be measurable, as some you would notice. Would it be worth the effort? On my side the answer is no.
If you look carefully, you will notice that in the log plot the value of growth is negative at the initial time, time = 0. Starting a completely new project can be overwhelming due to the depth of the subject, sometimes leaving you feeling as though you know even less than nothing. The log curve illustrates this well, showing growth = 0 only at time = 1.
4. Exponential Function.
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The opposite of logarithmic growth is exponential growth. It starts slow but then speeds up, beyond imagination 3.
Do you recall how, during the early days of COVID, epidemiologists warned about the danger of infections growing exponentially? Unfortunately, this risk materialized in some countries, leading to an exponential demand for hospital care and, ultimately, numerous fatalities when the needed care could not be provided.
The biological transmission mechanism that enables viruses to spread and infect new hosts is successfully mimicked in digital world. In the later, the mechanism is responsible for spreading viral videos, news and scams. Same as people cough and sneeze at each other in vivo, people sneeze-share in silico, that effectively passes viral information to new internet users 4. Both types of infections should plateau at some point, because both are limited by the size of population. But until that time – hand-free ride.
Many finance-related phenomena have characteristics of exponential growth too: capital markets, savings, inflation. The famous compounding is nothing else than exponential growth of money over prolonged time, provided that you do not consume the earned interests/capital gain. There is a quote attributed to Einstein on compound interest being the 8th wonder of the world. “Who understands it, earns it; who doesn’t, pays it”, the quote goes further. I think it is not completely true, because we all pay it, even if we understand it. How? In a form of inflation. I guess after living through 2020s’ everyone knows that inflation is loss of purchase power. While we all experienced it on a relatively short time scale, it is more interesting to look at this phenomenon on a longer scale; Let’s see how cumulative inflation causes loss of purchasing power over decades.
Today, you need to spend 34 $ on goods from inflation basket to have the same purchasing power as 1 $ from 1913. It is 34x more! But it averages to “only” 3.17% yearly inflation, in 1913-2026. In other words, 100 $ invested in 1913, would need to earn at least 3.17% yearly, to keep its purchasing power today, and grow to modest 3400 $. 100*1,0317113, gain to the power of time! As I read from my earlier work, average geometric return of American stock market in 113-year-long period of time 5 is 7.2%. If investing in 1911 was as simple as it is today, 100 $ would have grown to… 260 000 $. No typo, two hundred sixty thousands, that is.
5. Sigmoid Function.
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There is one more function I want to talk about: a sigmoid function. All-or-nothing. The change comes unannounced, and once it happens it is complete. Nowadays it is a very hot curve, as it is no.1 activation function used in neural networks architectures. For me, it is a titration curve 6. That curve nearly caused me to repeat analytical chemistry lab. Frist and foremost, I’m glad I haven’t touched a burette since. Secondly, I learned a valuable lesson: there are processes in which nothing happens for most of the time, and then the whole transition occurs in a very narrow range of time. If you miss a tiny bit the time when the change happened, you will be way off from the midpoint on the y-scale.
Closing Remarks
All the plots have small inset depicting cumulative growth at certain time value, entitled AUC - area under curve. The AUC is expressed in percentages, to make it easy to compare between the plots. Just have a look how it changes if you select different time ranges with the slider. You can adjust both ends of the range!
I find it handy to have these functions in the back of my mind, when encountering changes and thinking how they might develop. One of the mental shortcuts I use to simplify the complex reality. Feel free to use it too. And stop abusing linear function. Or just ignore all what I wrote: Pau Casals, probably the greatest cellist of the XX century, when asked in his late 80’ why is he still practicing 4h a day he said:
Because I am still making progress.
Thank you for reading.
Footnotes
I am not saying the analyses are useless, I am saying they are useless as prognoses. They might be of value for exploratory purposes, if you want to challenge your private crystal bowl predictions.↩︎
Market is entering bearish territory if it is -20% from its max price.↩︎
Initially I wanted to have same range for all functions presented in this post. I started with 0-10, but quickly realized I need to expand to 20, to capture plateauing of logarithmic curve. Using this range for exponential function, was, ekhm, challenging to apply. At t=20, growth(20) ≈ 485 million. Given growth = 150 requires 10 cm on the laptop screen, growth = 485 000 000 would need… more than 320 km to be ploted. Let’s call it infinite scroll.↩︎
IT would be completely speechless without biology.↩︎
The used range of market returns is shifted by 2 years, due to data availability, so it covers years 1911-2024.↩︎
Titration is a tedious exercise all chemistry adepts have to do at some point in their training. It has to do with quantification of a substance in solution. Imagine you are in a kitchen and you find an old bottle of vinegar. You know it is vinegar, because you can smell it, but you don’t know what is its concentration, because the label felt of (yes, it is safe to use it. It is not spoiled: acetic acid is already an oxidated substance, so that there is no easy way to be more spoiled. Wine can turn into vinegar, but not otherwise. Turning vinegar into wine, that would be a real miracle). To learn the concentration, you could perform a titration exercise: that is to let the vinegar, which is an acid, to be neutralized by a base. Neutralization needs to happen stepwise, so that you can capture the equivalence point, a moment when acid is neutralized by base. This point it is in the middle of the steep slope. To give you any chance to succeed, you do the exercise in presence of an indicator – a substance that changes color when the acid is neutralized. The trick is to capture the very moment of color change. To pass an exercise you give your readout from x-axis, that corresponds to consumption of base. Being tiny bit off on x-axis translates to being faaaar off on y-axis, what I had a chance to learn the hard way, multiple times.↩︎