How convincing is the Bitcoin Powerlaw?
By: The BitLog Team | Last update: Aug. 2025 | 2 min read
"Extraordinary claims require extraordinary evidence."
-Carl Sagan
The fact that bitcoin's price follows a powerlaw in time is a surprising claim. To appreciate this, let's first look at other known assets, specifically the S&P 500 economic index.
Like many assets based on population and technological growth, the S&P 500 grows roughly exponentially, and has done so for over 6 decades. When we plot using a linear scale in time and a logarithmic scale in price, it appears as a straight line (If you haven't done so, please look at the "Why Logarithmic Scale" article).
We also compare to both linear and powerlaw trends passing through the same beginning and end points. Recall that exponential growth and powerlaws are not the same!
We also include the option of comparing the rise of the S&P 500 to the price of bitcoin.
The S&P 500 fit to an exponential trendline is good but not great - though it's certainly a better fit than powerlaw or linear trends!
The point is that it doesn't have to be an excellent fit in order to be convincing, because roughly exponential growth of the S&P 500 index is perfectly reasonable.
Let's return to bitcoin
We will compare the bitcoin price to its lower bound powerlaw trendline and include two more trendlines; linear and exponential.
The difference is stark. The growth of bitcoin is nowhere near an exponential or linear trend! Despite the large volatility, it's clear that the powerlaw provides a much better description for the price.
Bitcoin seems to have followed this simple trend for 15 years despite regulation, crypto fraud, wars, and a pandemic.
The reason that the difference between types of trends above is noticeable has to do with the very large (million-fold!) increase in bitcoin price. The larger the price increase while following the trendline, the more convincing the powerlaw behavior.
As a counter example, let's zoom in to only the last five years of bitcoin price and make the same comparison. Since the price here only increases by a factor of 10, the difference between exponential and powerlaw trends is not as pronounced, and the correct choice of trend, if one exists, is not immediately apparent.
So is it convincing?
Your call.
We believe there is good evidence for powerlaw behavior.
This powerlaw fit is also simple, and contains only two free parameters (the coefficient and the power), which is more convincing than a fit to a very complex function.
Why this behavior occurs, if an explanation exist at all, is a question we address in the article "Why is BTC a Powerlaw?".