Bitcoin Power Law model / logarithmic regression bands (AUD)

Historical sigma extremes

Bitcoin's log-regression history is best understood through its extreme deviations. The table below shows the months where Bitcoin printed furthest above and furthest below the regression fair-value line.

Note: the 2013 and early-2014 readings are the most extreme in the data, and January 2013 is simply the first month of the series. Early months sit at the short end of the log-time axis, where a few data points have a large effect on the fit, so read them with caution. The 2017 top at +2.32σ is the most extreme reading from a mature market.

Where does Bitcoin currently sit on the log regression?

As of the most recent monthly close (October 2026), Bitcoin sits at -0.75σ from the long-run fair-value line, which is below fair value. The latest AUD price is A$118,282 against a regression fair-value estimate of A$184,605 for that month, a 36 percent discount. The central -0.5σ to +0.5σ band has held 35 percent of monthly closes.

For cycle-positioning context: the 2013, 2017 and 2021 tops printed above +1.5σ, but the 2024-2025 peak only reached +0.75σ. At the lows, 2015 and 2022 went below -1σ (-1.72σ and -1.19σ), while December 2018 (-0.70σ) and March 2020 (-0.63σ) bottomed closer to the line. The latest decline reached -1.14σ in June 2026.

Has the log regression predicted Bitcoin cycle tops?

Partly. The 2013 and 2017 cycle tops printed above the +2 sigma band (+3.38σ in November 2013, +2.32σ in December 2017). The 2021 cycle peaked at +1.97σ (March 2021), just inside the +2σ envelope, and the 2024-2025 cycle at only +0.75σ. Timing within the upper bands varied, and each cycle's peak has come in closer to the regression line than the last.

The log regression identifies historically rare zones, not specific peak dates. It is most useful as a confirmation indicator alongside other cycle tools (Pi Cycle Top, Mayer Multiple, Risk Metric).

What is logarithmic regression?

Bitcoin's price has appreciated by roughly six orders of magnitude since 2010 (from cents to over 100,000 USD). A linear chart of this history is unreadable: the early years compress to a flat line at zero, and recent moves look like vertical spikes. The fix is the logarithmic transformation.

Transform both axes: take the log10 of price (so 10x price moves become equal vertical distances) and take the log10 of time since some reference date (so the curve's growth slows linearly). In this transformed space, Bitcoin's long-run growth becomes nearly a straight line. That straight line is the logarithmic regression.

The mathematical interpretation is that Bitcoin's price has grown according to a power law in time: price ≈ k × t^n, where t is days since the genesis block and n is the slope of the regression line. The intercept k is determined by where on the line Bitcoin started. Both parameters are estimated from the fit.

The slope n depends on the data window and currency. On the AUD monthly data from January 2013 it is 5.41, so Bitcoin's AUD price has grown roughly in proportion to time since the genesis block raised to the power 5.41, a far faster growth rate than any traditional asset class.

The Bitcoin Power Law (Santostasi model)

The expression price ≈ A × t^n introduced in the previous section is the definition of a power law. When a relationship is a power law, plotting it in log-log space produces a straight line. The slope of that straight line is the exponent n, and the intercept is log(A). The fitted log regression on this chart is, mathematically, the Bitcoin Power Law model. The terminology differs but the maths is identical.

The power-law framing was formalised by Giovanni Santostasi, a physicist, in his 2024 paper "The Bitcoin Power Law Theory" (working paper, Burger had reached the same empirical fit earlier in 2019, and the Trolololo poster on Bitcointalk in 2014 had it earliest of all). Santostasi proposes a physical interpretation rather than just an empirical observation:

• Network effects (Metcalfe-style). Bitcoin's user base grows over time. The value of a network scales super-linearly with the number of participants. Combine these two and price grows as a power of time.
• Supply discipline. Bitcoin's issuance schedule is deterministic and asymptotically zero (capped at 21 million). New supply does not respond to demand. This produces an asymmetric supply curve that amplifies demand-driven price moves.
• Result. Network-adoption growth times supply-asymmetry produces a roughly exponential demand curve against a near-fixed supply curve, which in log-log space is a straight line - i.e., a power law.

The empirical AUD-native fit on this chart returns an exponent of n ≈ 5.41. Santostasi's published USD fit is about 5.8. The gap comes from the shorter AUD data window (from 2013 rather than 2010) and the AUD-USD exchange rate trajectory, so compare the two with care.

What's new in the framing. Calling the model a "logarithmic regression" describes the statistical method. Calling it a "power law" describes the resulting functional form and connects Bitcoin to a wide family of natural phenomena (city populations, earthquake magnitudes, word frequency distributions, network topology) that also follow power laws. The connection matters for two reasons: (1) it provides a mechanistic explanation (network effects + supply discipline) for why Bitcoin behaves this way, rather than just an empirical curve fit; (2) it predicts the exponent should stay roughly stable as long as the network-adoption + supply-discipline mechanism continues to operate. The second prediction is the one to watch: the last two cycle peaks have come in well below the historical upper bands.

Citation. Giovanni Santostasi, "The Bitcoin Power Law Theory", working paper, 2024. Distributed on Twitter/X as @Giovann35084111 and publicly archived. Earlier prior art: Harold Christopher Burger, "Bitcoin's Natural Long-Term Power-Law Corridor of Growth" (hcburger.com, 2019). Original Bitcointalk thread by user "Trolololo" (2014, link archived; original post: bitcointalk.org/index.php?topic=831547.0).

How to read the bands

Five visual elements:

• Central regression line (solid gold). The long-run fair value of Bitcoin in AUD at each point in time. The slope of this line is the long-run growth rate.
• ±1σ bands (dashed yellow/green). One standard deviation above and below the central line. 66 percent of historical monthly closes have been within this band. When Bitcoin is between ±1σ, it is in the historical "normal" range.
• ±2σ bands (dashed red/dark green). Two standard deviations above and below. 98 percent of monthly closes are within this band. The +2σ zone is the historical "overheated" band; the -2σ zone is the historical "undervalued" band.
• Gold dots. Each dot is a monthly close. Their distribution shows where Bitcoin has actually traded relative to the regression line.
• White circle. The most recent monthly close. Shows where Bitcoin sits today on the regression bands.

Practical interpretation:

• Bitcoin above +2σ: historically rare (4 months in the data). Reached around the 2013 and 2017 tops; not reached in 2021 or 2024-2025.
• Bitcoin below -2σ: never reached on the AUD fit. The deepest months were September 2015 (-1.72σ) and December 2022 (-1.19σ).
• Bitcoin near the central line: historical centre of the cycle range. No strong directional bias from the model.
• The model identifies HISTORICALLY RARE ZONES, not specific peak or bottom dates. Visits to the band edges have varied in duration across cycles.

Key Bitcoin events on the chart

The chart annotates 12 major Bitcoin events as vertical markers with abbreviated tickers above each line. Hover any marker for the full event name, date, and context. Toggle the annotations on or off using the "Hide events / Show events" button in the chart toolbar.

Marker colours: red = negative shock or cycle top; green = positive catalyst; gold = halving / cycle structural marker.

Methodology

The regression is computed with standard ordinary least squares on the transformed coordinates:

1. For each monthly close in AUD, compute (log10(days since Bitcoin genesis), log10(price in AUD)).
2. Fit a linear regression: log10(price) = slope × log10(days) + intercept.
3. Compute the residual: residual = actual log10(price) - predicted log10(price).
4. The standard deviation of the residuals is σ. The ±1σ and ±2σ bands are the central regression line offset by ±σ and ±2σ in log-price space.
5. Re-fit on every page load to incorporate the latest monthly close.

The Bitcoin genesis block timestamp (3 January 2009) is the t=0 reference. Days since genesis is the time axis. AUD monthly closes are the last daily close of each month from January 2013: Bitstamp BTC/USD converted at the daily AUD/USD rate, with Kraken's native BTC/AUD market for the last two years. Refreshed automatically every day.

The current fit parameters (slope and σ) are displayed in the meta strip directly under the chart. They update as each new month is added.

Where the model breaks down

Logarithmic regression is a statistical pattern-recognition tool, not a fundamental valuation framework. It has several known limitations:

• It assumes Bitcoin's growth rate is stable over time. The regression slope is constant in the model. If Bitcoin's adoption curve shifts (faster or slower than the historical fit), the model under- or over-projects.
• It is a backward-fitted model. The slope and intercept are fitted to past data. The model performs best on data the fit was trained on; out-of-sample predictions carry larger uncertainty.
• The bands are statistical, not causal. Visiting +2σ does not cause a cycle top; it is statistically associated with cycle tops in the historical sample.
• The model assumes Bitcoin continues to follow a power-law trajectory. If Bitcoin's long-run growth transitions to a different functional form (linear, S-curve, mean-reverting), the log regression bands will eventually disagree with future price action.
• Small data windows produce unstable fits. Refitting the model only on recent data (e.g., 3-year window) produces wildly different slopes than the full-history fit. The slope reported on this chart uses the full available history.

Despite these limitations, the framework has tracked Bitcoin's long-run path across several cycles and has become a widely-referenced chart in the long-term-investor community, even as recent peaks have fallen short of the upper bands. The chart above is the AUD-native version, recomputed on every page load.

Related tools

• Bitcoin Rainbow Chart (AUD) - same regression math, 9 sentiment-labelled bands (Fire Sale to Maximum Bubble Territory) for at-a-glance cycle positioning.
• Bitcoin Monthly Returns Heatmap (AUD) - month-by-month colour-coded returns. Complements the log regression by showing the volatility around the trend line.
• Crypto CGT Calculator - apply the ATO 50 percent discount to a specific Bitcoin disposal.
• Tax-Loss Harvesting Calculator - estimate EOFY tax savings from realising losses.
• Best Crypto Exchanges Australia 2026 - AUSTRAC-registered exchanges to buy and hold Bitcoin.

Frequently asked questions

About the author

Govind Satoshi

Former Institutional Trader. Founder, SatoshiMacro.

Traded allocated institutional capital at a Sydney proprietary trading firm.