Long-term power law
Long Term Power Law
Log price over log time — the straight line bitcoin has grown along.
Log price against log time since genesis — growth has hugged a straight power law for its whole life. The corridor is the fit ± one and two standard deviations. A regularity, not a promise.
How this is built
Log price over log time — the straight line bitcoin has grown along.
Formulalog₁₀ P = a + b · log₁₀(t − t₀)
AuthorGiovanni Santostasi
CoinGeckoPrice, market caps, all-time highs and dominance.
Blockchain.comFull-history market price used to backfill the long charts.
The indicator is Giovanni Santostasi's published idea, and Perly claims no authorship of it. Perly draws the chart from the data of the providers listed above, and that data remains theirs.
How to read the power law chart
Both axes are logarithmic. Up the side is bitcoin's price; along the bottom is time, counted in days since 9 January 2009 and stretched so that equal ratios of time take equal widths. On those axes a power law is a straight line, and that is the point of the chart.
The white line is bitcoin's closing price. The dashed violet line is the fitted power law, the trend. The dashed red and green lines above and below it are one standard deviation from the trend, the thin lines two, and the grey band between the thin lines is the ±2σ corridor. Hovering any day gives the close, the trend's value and the price as a multiple of the trend; the readout above gives today's.
Dashed verticals mark the halvings, and each series and the corridor can be switched off in the legend. There are no range chips on this chart; on a computer, drag across the plot to zoom and double-click to reset.
Distance from the trend at every cycle top and low
| When | BTC price | Trend | From trend |
|---|---|---|---|
| Cycle top10 Jun 2011 | $33.80 | $2.04 | +4.1σ |
| Cycle low21 Nov 2011 | $2.30 | $5.28 | −1.2σ |
| Cycle top04 Dec 2013 | $1,137 | $106 | +3.4σ |
| Cycle low14 Jan 2015 | $172 | $331 | −0.9σ |
| Cycle top16 Dec 2017 | $19,280 | $3,016 | +2.7σ |
| Cycle low15 Dec 2018 | $3,232 | $5,442 | −0.8σ |
| Cycle top08 Nov 2021 | $67,562 | $22,723 | +1.6σ |
| Cycle low21 Nov 2022 | $15,760 | $35,030 | −1.2σ |
| High so far06 Oct 2025 | $124,740 | $100,235 | +0.3σ |
| Low so far30 Jun 2026 | $58,566 | $127,231 | −1.1σ |
| Latest20 Oct 2026 | $81,558 | $140,202 | −0.8σ |
How Perly fits the power law
The fit is ordinary least squares of the base-10 logarithm of every daily close against the natural logarithm of the days since 9 January 2009, the day bitcoin's software was released and its first block after the genesis block was mined. It gives a line log₁₀(price) = a + b · ln(days). The standard deviation is taken of every close's distance from that line, in the same base-10 logarithm, and the corridor is drawn at one and two of them either side.
The fit uses the whole record and is redone every time the page is rebuilt, so the line moves a little as each new close arrives.
Where it comes from
The observation that bitcoin's price follows a power law of time is Giovanni Santostasi's, and he is credited under the chart. Versions of the chart differ in the day they count from, the data they fit and how they draw the corridor, so the trend here will not match another site's to the dollar.
What the record shows
Measured against today's fit, the tops of 2011, 2013, 2017 and 2021 each stood fewer standard deviations above the trend than the one before, and the cycle lows have sat between three quarters of one and a little over one standard deviation below it. The table gives the figures.
A straight line on log-log axes is a strong regularity across a young asset's whole history, but it is a description. Nothing in it says the years ahead must follow.
Questions about the bitcoin power law
What is the bitcoin power law?
The observation that bitcoin's price has grown roughly as a power of the time since its launch: on a chart with logarithmic price and logarithmic time, the history lies close to a straight line. This chart fits that line to every daily close and draws a corridor around it.
What is the power law price of bitcoin today?
The readout above the chart gives the trend's value for today, and the table's last row gives the latest price's distance from it in standard deviations.
How is the power law different from the rainbow chart?
The rainbow fits a logarithmic curve against ordinary time and hangs bands of fixed width on it. The power law fits a straight line on log-log axes and draws its corridor at the fit's own standard deviations. They are separate fits and do not agree exactly.
Who came up with the bitcoin power law?
Giovanni Santostasi, who is credited under the chart. Perly's fit is its own, computed from its daily record.
Why does the power law count days from 9 January 2009?
It is the day bitcoin's software was released and its first block after the genesis block was mined. A power law needs a day to count time from, and different versions of the chart choose different ones.
Indicators are descriptive statistics about the past, not predictions. Nothing on this page is financial advice.
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