Logarithmic regression and rainbow charts

Logarithmic regression and the rainbow chart are the pre-Power-Law family of long-term Bitcoin price frameworks. The basic logarithmic regression fits Bitcoin's price against the logarithm of time (or against time itself, with log-transformed price), producing a slowly-curving long-term trend; the rainbow-chart visualization, developed by Bitcoin Reddit user "Trolololo" beginning around 2014, overlays the trend with color-coded bands describing cycle position (from blue "fire sale" through green "accumulate" up to red "maximum bubble territory"). The frameworks are functionally subsumed by The Power Law model — the Power Law being a refined log-regression with rigorous mathematical form and causal mechanism — but they retain historical importance as the framework that anchored long-term Bitcoin thinking before the Power Law's 2018-2019 emergence, and continued practical utility as accessible cycle-positioning visualizations. The rainbow chart specifically remains widely shared in Bitcoin discourse.

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Why this note matters

The log-regression and rainbow-chart frameworks are the immediate precursors to the Power Law model and remain widely cited in Bitcoin discourse. Three reasons they’re worth a dedicated note:

1. Historical importance: Before Santostasi and Perrenod formalized the Power Law in 2018-2019, log-regression frameworks were the dominant long-term price-trajectory tools. Understanding the contemporary frameworks requires understanding what they replaced.
2. Continued practical use: The rainbow chart remains widely shared on Bitcoin social media and in analyst publications as a simple cycle-positioning heuristic. Engaging the framework substantively, including its limitations, is part of honest treatment of the discourse.
3. The relationship to Power Law clarifies both frameworks: The Power Law is, in effect, a rigorous version of the log-regression intuition. Treating the two together highlights what was right in the earlier framework (the rough log-linear trend) and what the Power Law added (specific functional form, causal mechanism, mathematical-physical grounding).

The note exists to (1) present log-regression substantively, (2) describe the rainbow-chart visualization and its development, (3) honestly characterize the framework as the Power Law precursor, (4) engage where it retains pedagogical value, and (5) handle the counter-arguments — particularly that the framework has been superseded.

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The logarithmic regression framework

Bitcoin’s price plotted on a linear-vs-linear chart is dominated by recent prices — the early-history (sub-$1) prices are invisible at any scale that shows current prices. The standard analytical move is to plot log price against linear time (or log price against log time), producing a much more interpretable picture of Bitcoin’s history.

The basic log-regression form:

$\log (P)=a+b\cdot t$

or equivalently:

$P=e^a\cdot e^{bt}$

This is exponential growth — price doubles every $\ln (2)/b$ time units. Fits of Bitcoin’s price history to this form give doubling times in the range of 8-12 months historically, though the doubling time has lengthened in later cycles.

The log-time variant (closer to Power Law):

$\log (P)=a+b\cdot \log (t)$

or equivalently:

$P=e^a\cdot t^b$

This is power-law growth in time — and is exactly the functional form the Power Law model formalizes. The log-time-regression framework is therefore a precursor of, and ultimately equivalent to, the Power Law model. The two were developed somewhat independently — log-regression came out of Bitcoin community practice; Power Law came out of physicist analytical traditions — but they describe the same mathematical relationship.

Where they differ: the Power Law treats the functional form ($P\propto t^n$) as theoretically motivated (network-effects compounding on adoption-curve growth) with a specific exponent fit empirically; log-regression treats the form as just an empirical curve-fit without strong theoretical motivation. The substantive content is similar; the analytical rigor is different.

Practical specifications

Various analysts have published log-regression fits to Bitcoin price across the years. Key parameters across different specifications:

• Slope (in log-time-regression form): typically 5-6, consistent with the Power Law exponent of ~5.7
• Intercept: varies with start-time choice (genesis block vs first market price vs first halving)
• R²: typically above 0.95 in log-log regressions across Bitcoin’s full history

The strong fit is the same finding the Power Law produces, expressed in different vocabulary. See The Power Law model for the rigorous treatment.

Distinction between log-price-on-time and log-price-on-log-time

Two specifications are sometimes conflated and should be distinguished:

• Log price on linear time (exponential growth) — fits Bitcoin’s early years but progressively diverges from later-cycle data; predicts unrealistic appreciation in the long run
• Log price on log time (power-law growth) — fits Bitcoin’s full history substantially better and is the form the Power Law model uses

Early log-regression analyses (2013-2016) typically used the linear-time variant; later analyses moved to the log-time variant. The rainbow-chart frameworks have used both.

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The rainbow chart

The Bitcoin rainbow chart is a specific visualization developed by Reddit user “Trolololo” (also written as “Trololo”) beginning around 2014. The chart:

1. Plots Bitcoin’s price history on a logarithmic vertical axis against linear time
2. Overlays a logarithmic regression trend line
3. Adds color-coded bands above and below the trend representing cycle-position descriptors

The standard rainbow bands

The bands are described with semi-humorous cycle-sentiment labels:

Band (top to bottom)	Color	Sentiment label
Top band	Red	Maximum bubble territory — “Sell. Seriously, SELL!”
Above top trend	Orange-red	Sell. Seriously, SELL!
Top of trend	Orange	FOMO intensifies
Mid-upper	Yellow	Is this a bubble?
Middle	Yellow-green	HODL
Mid-lower	Light green	Still cheap
Lower	Green	Accumulate
Lower middle	Cyan/blue-green	BUY!
Bottom	Blue	Basically a fire sale

The bands are essentially standard-deviation channels around the log-regression trend, with cycle-position-descriptive labels chosen for accessibility and humor.

Origins and evolution

The original Trolololo rainbow chart appeared on Reddit’s r/Bitcoin around 2014. The visualization spread through Bitcoin social media and was later picked up by mainstream Bitcoin charting sites (LookIntoBitcoin, Glassnode, various Bitcoin Magazine analyses). The specific band thresholds and label wording have evolved across implementations, but the core idea — logarithmic regression with color-coded cycle-position bands — has remained consistent.

Variants include:

• LookIntoBitcoin’s rainbow chart — with refined labels and band thresholds
• The “BlockchainCenter” rainbow chart — alternative implementation
• Hash-rate-based rainbow charts — applying the same visualization to hashrate rather than price

What the rainbow chart claims

The substantive claims of the rainbow chart:

1. Bitcoin’s price has a long-term trend (the regression line)
2. The trend is sufficiently stable that cycle-position can be described by distance from trend
3. Color-coded bands provide accessible visualization of cycle-position
4. Specific bands have empirical history — substantial portion of historical price action has remained within the bands; band-extremes have coincided with cycle peaks and bottoms

The third and fourth claims are the rainbow-chart’s specific contribution beyond plain log-regression. The rainbow chart is essentially a pedagogical visualization of the log-regression framework’s cycle-positioning utility.

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Empirical assessment

How well do log-regression and rainbow-chart frameworks perform empirically?

For log-regression specifically:

• Strong long-term fit ($R^2>0.95$ over Bitcoin’s full history) — same finding the Power Law produces
• Cycle-position utility — distance from the regression line has provided useful cycle-positioning information across all four observed cycles
• Diminishing returns naturally embedded — the log-time variant predicts cycle-by-cycle appreciation attenuation consistent with observed pattern

For the rainbow chart specifically:

• Visual accessibility — the chart has been very widely shared and is one of the most-cited Bitcoin visualizations in popular discourse
• Cycle-top warnings — the upper bands have lit up at every cycle peak (2013, 2017, 2021) before substantial drawdowns
• Cycle-bottom signals — the lower bands have lit up at every cycle bottom (2015, 2018, 2022) before substantial recoveries
• Mid-cycle ambiguity — within-cycle pricing has typically sat in middle bands where signals are weaker

Where it has been weaker:

• No causal mechanism — log-regression and rainbow-chart frameworks are descriptive without underlying theory
• Specific predictions are vague — the bands are wide; cycle-top and cycle-bottom timing predictions are uncertain
• Subsumed by Power Law — the Power Law provides rigorous version of the same intuition with better grounding
• The 2022 drawdown depth went below the bottom band in some implementations, which was unusual and the framework’s natural variability didn’t fully absorb

The honest reading: log-regression-style frameworks are pre-formal versions of the Power Law. They retain pedagogical value for accessibility but have been substantively superseded for serious analytical work.

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Relationship to the Power Law model

The log-regression framework and the Power Law model are effectively the same framework at different levels of formalization:

Dimension	Log-regression (rainbow chart)	Power Law model
Functional form	Log price = a + b log(t)	$P(t)=A\cdot t^n$
Mathematical identity	Same equation, different notation	Same equation, different notation
Theoretical motivation	Empirical curve-fit	Network-effects compounding on adoption-curve growth
Specific exponent	Empirically fitted, no derivation	$n\approx 5.7$, with mechanistic derivation (Santostasi-Perrenod 2026)
Cycle treatment	Bands as descriptive cycle-positioning	Power Law corridor as cycle-positioning, with log-periodic structure as deeper framework
Causal account	None implicit	Adoption + Metcalfe-style network value
Status	Precursor framework, retained for accessibility	Current consensus rigorous framework
Cycle-mechanism explanation	Bands deviate at peaks/troughs; no causal account	Log-periodic oscillations (Log-periodic cycles and the Perrenod-Santostasi wave model)

The synthesis: log-regression / rainbow-chart frameworks captured the right empirical pattern (power-law-in-log-time growth) but didn’t have the theoretical apparatus to ground it. The Power Law model provides the missing theoretical structure. For accessible cycle-positioning visualization, the rainbow chart remains useful; for serious analytical work, the Power Law is the right framework.

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Implications and use

For practical use of these frameworks:

• As cycle-positioning visualization: the rainbow chart remains useful, particularly for newcomers and accessible-pedagogy contexts
• As a price-prediction tool: log-regression is largely superseded by Power Law; use Power Law for rigorous prediction work
• As an educational artifact: the rainbow chart’s pedagogical accessibility has substantive value even where its analytical content is limited
• As cycle-top warning: upper-band signals have empirically coincided with cycle peaks across cycles; useful directional signal even if specific timing is uncertain
• As cycle-bottom signal: lower-band signals have empirically coincided with cycle bottoms across cycles; useful for accumulation timing

The framework should be cited honestly: as a precursor that has been substantially superseded, but with continued pedagogical and visualization value.

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Counter-arguments and tensions

The framework is just curve-fitting

The argument: Logarithmic regression is statistical curve-fitting without underlying theory. Any sufficiently flexible curve can fit historical data; the lack of theoretical motivation means the framework has no real predictive content. The rainbow chart’s bands are arbitrary; the labels are humor masquerading as analysis.

Response: Substantively correct as a critique of log-regression as a standalone framework. The defense is that:

1. The empirical fit is real — Bitcoin’s price has tracked a log-linear-in-log-time trend since 2009 across multiple cycles, which is more than pure curve-fitting would predict
2. The successor (Power Law) provides the missing theoretical grounding — what log-regression lacked, Power Law supplies
3. The framework’s predictive utility has been substantive — cycle-position signals have empirically tracked actual cycle dynamics

The honest reading: log-regression is curve-fitting with retrospective theoretical motivation. The Power Law makes the motivation explicit and rigorous.

The rainbow chart’s labels are unscientific

The argument: Labels like “Maximum bubble territory” and “Basically a fire sale” are not analytical claims; they are humor or sentiment-anchored language that anchors expectations in problematic ways. Investors who treat the rainbow chart as analytical may make decisions based on what is essentially marketing.

Response: Fair as a critique of presentation. The defense is that the rainbow chart is explicitly pedagogical/accessible rather than rigorously analytical — the labels are descriptive sentiment-anchors, not predictions. Sophisticated users understand the framework’s limitations; less-sophisticated users get a visualization that’s at least directionally informative. Concerns about misinterpretation are real but not unique to rainbow charts.

Subsumed by Power Law

The argument: The Power Law model provides everything log-regression provides, plus theoretical grounding, plus rigorous parameter specification, plus integration with broader network-economics frameworks. There is no analytical reason to use log-regression rather than Power Law for serious work. The framework is essentially obsolete.

Response: Largely correct. The defense is that the rainbow chart retains pedagogical value the Power Law doesn’t replace — the visualization is more accessible than the Power Law’s mathematical formulation. For analytical work, Power Law; for educational and accessible-visualization work, the rainbow chart still has a role. The framework is supplemental, not analytically primary.

The bands are not well-calibrated

The argument: Different rainbow-chart implementations use different band thresholds; there’s no canonical “correct” version. The lack of calibration means cycle-position descriptors are framework-specific rather than analytically robust.

Response: Fair. Standardization has been weak. The most-used implementations (Trolololo, LookIntoBitcoin, BlockchainCenter) differ in band thresholds, color choices, and label wording. A user looking for cycle-positioning should use multiple implementations or the underlying Power Law corridor (which is calibrated to standard-deviation channels around the trend) rather than relying on a single rainbow-chart variant.

Log-regression overstates predictability

The argument: Treating Bitcoin’s price as following a deterministic log-regression trend understates the actual variance in cycle dynamics. The 2022 drawdown to ~$15.5K went below most rainbow-chart bottom bands; the post-2024 cycle has produced unusual dynamics. The frameworks suggest more predictability than the data supports.

Response: Real concern. The frameworks should be cited with appropriate uncertainty bands rather than as deterministic predictions. The Power Law corridor (standard-deviation channels) handles this more honestly than the rainbow chart’s relatively narrow bands.

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Open questions for further development

• Should the framework be retired in favor of Power Law for all serious use, or does the rainbow chart retain enough pedagogical value to keep it in the analytical toolkit?
• How should the rainbow-chart bands be calibrated as Bitcoin matures? Cycle-amplitude attenuation may require evolving band thresholds.
• Is there a useful integration of log-regression visualization with on-chain cycle indicators? Combining frameworks might produce better cycle-positioning than either alone.
• What is the appropriate response to rainbow-chart-anchored expectations in popular discourse? The framework’s limitations are real; cultural anchoring on its predictions can be harmful.
• How does the framework engage Layer 2 and institutional dynamics that may shift Bitcoin’s structural price-formation? The base-layer regression may need updating as Bitcoin’s architecture evolves.

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Canonical sources for this note

Foundational rainbow-chart sources

• “Trolololo” Reddit posts (r/Bitcoin) circa 2014 — original rainbow-chart development
• LookIntoBitcoin rainbow-chart implementation (lookintobitcoin.com)
• BlockchainCenter Bitcoin Rainbow Chart (blockchaincenter.net)

Historical log-regression analyses

• Various Bitcoin Magazine and Bitcoin Talk forum analyses through 2014-2018 — early log-regression frameworks
• Various BitcoinPro and analyst-publication log-regression fits

Power Law as successor

• Giovanni Santostasi, “The Bitcoin Power Law Theory” — the framework that supersedes log-regression
• Stephen Perrenod, various Substack writings
• Santostasi and Perrenod, mechanistic-derivation Scientific Bitcoin Institute paper

Cycle-positioning visualization adjacent

• Various Glassnode, Checkonchain, and Coin Metrics cycle-positioning frameworks
• Power Law corridor implementations (bitcoinpower.law, various analyst tools)

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Related notes

• The Power Law model — the successor framework that supersedes log-regression
• Stock-to-flow model — alternative framework engaged with the same trajectory question
• Adoption curves — adoption-side framework grounding the Power Law mechanism
• Metcalfe’s Law applied to Bitcoin — network-value framework grounding the Power Law mechanism
• Four-year halving cycles — cyclical framework that operates alongside the trend
• Lindy effect and Bitcoin — survival-based framework adjacent to long-term trajectory
• Diminishing returns thesis — cycle-attenuation framework naturally embedded in log-regression
• Long-term price models and cycles — sub-MOC for the price-models area
• Giovanni Santostasi — Power Law originator
• Stephen Perrenod — Power Law co-developer
• Plan B — Stock-to-flow framework; engaged critically
• Dylan LeClair — market analyst engaging cycle-positioning frameworks