The Bitcoin Power Law is the claim that price follows the simple relationship price = a · t^b, where t is the number of days since the Bitcoin genesis block (January 3, 2009), a is a scale coefficient, and b is the power exponent (empirically around 5.8). This page explains why that formula produces the chart you see and how its numbers are found.
Why the curve becomes a straight line
On an ordinary chart, a · t^b is a steep upward curve. But take the logarithm of both sides and something clean happens:
log(price) = log(a) + b · log(t)
That is the equation of a straight line, y = mx + c, where y = log(price), x = log(t), the slope m is the exponent b, and the intercept c is log(a). So when both axes are logarithmic (a "log-log" chart), any true power law appears as a perfectly straight diagonal line. Bitcoin's price, plotted this way across more than a decade, hugs such a line remarkably closely - which is the whole reason the model is taken seriously.
Fitting a and b
Because the log-log form is linear, the coefficients are found with ordinary least-squares linear regression on the logarithms. You take every historical price point, convert to (log t, log price), and find the straight line that minimizes the squared vertical distance to those points. The slope of that line is b; raising e to the intercept gives a. On the interactive chart, the Best fit button runs exactly this calculation, and the R-squared readout reports how much of the variation the line explains (historically well above 90%).
Where the bands come from
Price never sits exactly on the line - it oscillates above and below. To measure that scatter, we look at each point's residual: how far its log-price sits from the fitted line. The standard deviation of those residuals, called sigma (σ), summarizes the typical wandering. The bands are then drawn at trend × e^(±kσ). A band at ±1σ contains roughly 68% of history; ±2σ contains about 95%. The lower ±2σ edge is what the site labels the floor support, and the upper edge the ceiling.
What the math cannot tell you
Regression describes the past; it does not guarantee the future. A high R-squared means the line has fit well so far, not that the relationship is a law of nature. The exponent and the bands also shift as new data arrives. Treat the model as a disciplined way to ask "is today unusual?" - not as a forecast.
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Educational content. Not financial advice.