Bitcoin Power Law model / logarithmic regression bands (AUD)
The long-running logarithmic regression of Bitcoin's price against time - mathematically equivalent to Giovanni Santostasi's Bitcoin Power Law model. The central log-log fit is the power law trajectory price ≈ A × t^n; the standard-deviation bands around it define cycle-top and cycle-bottom zones. Used by long-term Bitcoin investors to identify overheated (+2σ) and undervalued (-2σ) regimes. AUD-native (most equivalent charts online are USD-only). Updated automatically on every site refresh.
Chart
Each gold dot is a Bitcoin AUD monthly close. The central gold line is the long-run logarithmic regression of price against time. The dashed lines above and below are ±1σ and ±2σ bands. The white circle marks the most recent month. Hover any point on the chart to see the exact price, fair value, and sigma deviation for that month.
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.
| Rank | Month | Sigma deviation | BTC AUD price | Cycle context |
|---|---|---|---|---|
| 1 | December 2017 | +2.50σ | A$18,377 | 2017 ICO-era cycle top |
| 2 | November 2017 | +2.02σ | A$13,247 | Pre-top mania month |
| 3 | March 2021 | +1.94σ | A$78,400 | 2021 first cycle peak |
| ... | centre band (fair value, -0.5σ to +0.5σ) | |||
| -3 | December 2022 | -1.45σ | A$23,986 | FTX-collapse cycle low |
| -2 | August 2015 | -1.62σ | A$307 | Post-Mt-Gox bear bottom |
| -1 | September 2015 | -1.71σ | A$313 | Deepest historical undervaluation |
Note: early-2014 outliers (January 2014 at +2.96σ, February 2014 at +2.16σ) are partly an artefact of the regression fit on a short early sample where the slope was less constrained. Read with caution.
Where does Bitcoin currently sit on the log regression?
As of the most recent monthly close, Bitcoin sits at +0.37 sigma above the long-run fair-value line. That places it in the "fair value" classification band (-0.5σ to +0.5σ), which has contained roughly 38 percent of historical monthly closes. The current AUD price is A$207,385 against a regression fair-value estimate of A$167,679 - a 24 percent premium to fair value, but well inside historical volatility for the band.
For cycle-positioning context: every prior cycle top (2013, 2017, 2021) printed above +1.5σ. Every prior cycle bottom (Dec 2018, Mar 2020, Dec 2022) printed below -1.0σ. The current +0.37σ reading suggests Bitcoin is roughly mid-cycle, with meaningful upside before historical "overheated" territory.
Has the log regression predicted Bitcoin cycle tops?
Approximately. The 2013, 2017, and 2021 cycle tops all printed at or above the +2 sigma band. December 2017 (+2.50σ) and November 2017 (+2.02σ) were the most extreme readings of any post-2014 cycle. The 2021 cycle peaked at +1.94σ (March 2021), slightly inside the +2σ envelope. Caveats: timing within the band varied (BTC spent 2-4 months above +1.5σ in each cycle before the actual peak day), and the absolute price level at each top has trended higher in line with the regression line itself.
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 is roughly 5.5 to 6 over Bitcoin's full history, depending on the data window. That means Bitcoin's price has roughly grown with the 5.5th to 6th power of time since 2010, which is 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 approximately n ≈ 5.7. Santostasi's USD fit returns approximately n ≈ 5.8. The two are nearly identical because they describe the same underlying phenomenon; the small difference is the AUD-USD exchange rate trajectory layered on top. The exponent has stayed remarkably stable across every refit since 2014, which is the empirical anchor for the power-law thesis.
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. Both predictions have held up empirically for a decade.
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. Roughly 68 percent of historical monthly close prices 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. Roughly 95 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. Has preceded cycle tops in 2013, 2017, and 2021 within months.
- Bitcoin below -2σ: historically rare. Has preceded cycle bottoms in 2015, 2019, and 2023 within months.
- 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.
| Date | Marker | Event | Why it matters |
|---|---|---|---|
| 2014-02-24 | GOX | Mt. Gox collapse | The largest crypto exchange of the era files for bankruptcy after losing ~850,000 BTC. Triggers the multi-year bear market through 2015. |
| 2016-07-09 | HV2 | 2nd Bitcoin halving | Block reward drops from 25 BTC to 12.5 BTC. Marks the start of the 2017 bull cycle. |
| 2017-12-17 | TOP | 2017 cycle top | BTC hits ~A$25,000 in the ICO-era mania top. Beginning of a 12-month bear market that bottoms in December 2018 around A$4,500. |
| 2020-03-12 | COV | COVID crash | BTC drops 50%+ in a single day alongside the global risk-asset selloff. With hindsight, a generational buy zone. |
| 2020-05-11 | HV3 | 3rd Bitcoin halving | Block reward drops from 12.5 BTC to 6.25 BTC. Triggers the 2020-2021 bull cycle. |
| 2021-04-14 | COIN | Coinbase IPO + April top | Coinbase direct-lists on NASDAQ at a $250B valuation; BTC tops near A$85k in the same week. First major-exchange IPO marks broad institutional acceptance. |
| 2021-05-19 | CHN | China mining ban | China cracks down on Bitcoin mining. Network hashrate temporarily collapses 50%; BTC mid-cycle dump. |
| 2021-11-08 | ATH | November 2021 cycle top | BTC prints ~A$94,000 cycle high. End of the 2020-2021 bull market. |
| 2022-05-12 | LUNA | Luna / Terra collapse | Luna and UST collapse to zero in a week. Triggers a cascade of crypto credit failures (Celsius, Three Arrows Capital, Voyager). |
| 2022-11-11 | FTX | FTX collapse | FTX files for bankruptcy after an 8-day liquidity run. Marks the cycle-low zone for BTC at ~A$23,000. |
| 2024-01-10 | ETF | Spot BTC ETF approval | SEC approves the first US spot Bitcoin ETFs (IBIT, FBTC, ARKB, others). ~A$50B inflows in the following 12 months. |
| 2024-04-19 | HV4 | 4th Bitcoin halving | Block reward drops from 6.25 BTC to 3.125 BTC. The current cycle is dated from this event. |
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:
- For each monthly close in AUD, compute (log10(days since Bitcoin genesis), log10(price in AUD)).
- Fit a linear regression: log10(price) = slope × log10(days) + intercept.
- Compute the residual: residual = actual log10(price) - predicted log10(price).
- 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.
- Re-fit on every site build 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 price data is sourced from a public market-data endpoint, downsampled from daily to monthly closes.
The current fit parameters (slope, intercept, and σ) are displayed in the meta strip directly under the chart. These update each build as the data window extends.
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 held up reasonably well across three full Bitcoin cycles (2013, 2017, 2021) and has become a widely-referenced reference chart in the long-term-investor community. The chart above is the AUD-native version, recomputed independently on every site build.
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
Bitcoin logarithmic regression is a long-term fair-value model that fits a straight line to Bitcoin's price-over-time relationship in log-log space (logarithm of price on the y-axis, logarithm of days since the genesis block on the x-axis). Because Bitcoin's price has historically grown roughly exponentially with time, the log-log transformation produces a near-linear relationship. The fitted line represents long-run fair value. Standard-deviation bands above and below the line define overheated and undervalued zones. The model has been used to identify cycle tops and bottoms since at least 2014, and is mathematically equivalent to Giovanni Santostasi's Bitcoin Power Law model (a straight line in log-log space is exactly the equation P(t) = A × t^n, which is the definition of a power law).
Yes, mathematically. A straight-line fit in log-log space is the definition of a power law: log(price) = n × log(t) + log(A) is algebraically identical to price = A × t^n. The log regression bands on this chart are the +1σ and +2σ envelopes of the same fit. Santostasi's 2024 'Bitcoin Power Law Theory' paper formalises the model and proposes a physical interpretation (Metcalfe-style network adoption × a deterministic supply curve produces a power-law price trajectory). Earlier work by Harold Christopher Burger (2019) and Trolololo on Bitcointalk (2014) reached the same empirical fit without the formal physics framing. The chart here is the AUD-native implementation of the same mathematical model.
The slope of the log-log fit IS the power-law exponent. The SatoshiMacro AUD-native fit returns a slope of approximately 5.7 over the full 2014-today window, meaning Bitcoin AUD price has grown roughly as the 5.7th power of time since the genesis block. Santostasi's USD fit returns approximately 5.8. The two are close (AUD vs USD differ only by the AUD/USD exchange rate trajectory, which is a small effect over decade horizons). The exponent has stayed remarkably stable across all post-2014 refits, which is the empirical observation that motivates the power-law framing.
Bitcoin's price has appreciated by roughly six orders of magnitude since 2010 (cents to over 100,000 USD). A linear chart compresses early-history price action into a flat line and makes recent moves look like vertical spikes. A logarithmic transformation gives equal visual weight to percentage moves regardless of price level, which makes the long-run growth trend visible and the cycle bands meaningful. The same transformation is used for any asset class with multi-order-of-magnitude price history.
The bands are standard deviations of the regression residuals. The ±1σ band contains roughly 68 percent of historical monthly close prices; the ±2σ band contains roughly 95 percent. When Bitcoin trades above the +1σ band it is in the upper 16 percent of historical valuation relative to its long-run trend; above +2σ it is in the upper 2.5 percent. The bands are historically rare zones, not predictions. Bitcoin has visited each band several times in its history.
Approximately. Bitcoin's 2013, 2017, and 2021 cycle tops all occurred at or above the +2σ band, which is consistent with the framework's predictive value as a top-zone indicator. However, the timing of each top within the +2σ zone varied (some weeks to months above the band before the actual peak), and the absolute price level at each top has trended higher in line with the regression line itself. The model identifies historically rare zones, not specific peak dates.
Bitcoin is priced in USD on global exchanges, but Australian-resident investors measure portfolio value in AUD. The AUD-USD exchange rate moves independently of Bitcoin's price, which means the AUD-priced regression line has a slightly different slope than the USD-priced version due to AUD's long-term depreciation against USD. The AUD-native chart is the correct reference for an Australian-resident investor. Most equivalent charts online (Bitbo, Lookintobitcoin, Coinglass) are USD-only.
Standard ordinary-least-squares linear regression on the transformed coordinates. Each monthly close becomes a point at (log10(days since genesis), log10(price in AUD)). The fit minimises the sum of squared residuals on the log-price axis. Slope and intercept define the central regression line. The residual standard deviation defines the ±1σ and ±2σ band widths. Recomputed from scratch on every site build using the latest available data.
The stat strip directly under the chart shows Bitcoin's current sigma-deviation from the fair-value line, alongside the current price and the projected +2σ and -2σ levels. Positive deviations indicate Bitcoin is above the long-run trend; negative deviations indicate below. The interpretation paragraph translates the sigma-deviation into a plain-English assessment of the cycle phase.
The underlying data refreshes on every site build. The chart re-fits the regression and re-renders on every page load using the latest data. If the upstream source is unreachable during a build, the previous data set is preserved unchanged so the chart continues to render with the last-known-good data.
Twelve key events from 2014 to 2024: Mt. Gox collapse (Feb 2014), 2nd halving (Jul 2016), 2017 cycle top (Dec 2017), COVID crash (Mar 2020), 3rd halving (May 2020), Coinbase IPO + April top (Apr 2021), China mining ban (May 2021), November 2021 cycle top (Nov 2021), Luna / Terra collapse (May 2022), FTX collapse (Nov 2022), spot BTC ETF approval (Jan 2024), 4th halving (Apr 2024). Each event is plotted as a vertical dashed line with a coloured dot at the top: red for negative shocks or cycle tops, green for positive catalysts, gold for halvings and cycle-structural markers. Hover any marker for the full event description. Toggle the annotations on or off using the 'Hide events / Show events' button in the chart toolbar.
Halvings are deterministic supply events that occur every 210,000 blocks (approximately every 4 years). They reduce the BTC issuance rate by 50% and are widely understood to mark the structural start of each Bitcoin cycle. The price effect of a halving is gradual, not instant: BTC has typically rallied 12 to 18 months AFTER each halving, not on the day. Halvings are coloured gold (cycle-structural) rather than green (positive catalyst) because they're calendar-deterministic supply events, not unexpected news. The Bitcoin Halving Countdown tool shows the day-by-day countdown to the next halving (April 2028).
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