Oddbean new post about login | logout It should be identical, in practice, just scaled. The property of a log chart is that proportional changes have the same height. So 2x has the same vertical height whether it’s $0.01 to $0.02 or from $10k to $20k. This proportionality property exists irrespective of the base used in the log operation. The raw values will differ, but it’s common to scale the height of the chart based on the display anyway to focus on what you’re trying to show.
https://cdn.satellite.earth/c01c24ba7e849f37fb86d0f9825d659b7d48559d04ade1e8c27a80f81d83848d.png this is the comparison of the linear, log2 and log10 zero points on a graph obviously the log10 (which is implicitly what is meant when log is used alone) is a flatter curve the log2 curve decelerates slower and longer to reach the same gradient
https://cdn.satellite.earth/1ff6b87ed64a58d01af206edf1095d3909dd955d2ea1e4bafbf8ac0f29c7bed9.png this represents the two different zeroes based on log10 (black) and log2 (red) with their respective double and half values, which is the range you'd see on the chart i added this one to illustrate how especially in the early part of the graph the angle of the log2 version is going to mean a more linear result against time for our price chart
Yes, the log(2) curve will grow “faster”. Its values are scaled by a constant factor, namely log2(10) = 3.32. But when you render a chart for display, there’s an implicit scaling between the actual values and the position (in pixels) of that visual element (line). This implicit scaling factor arbitrary, and chosen by the charting library, or manually by the chart author, to showcase the phenomena under investigation. You could add to this chart another log line with base, say, 10 billion, and the other lines would be entirely squashed at the bottom. But the shape of each line would remain the same, preserving the change proportionality property.
yes and my whole point is currently what we see is a faster flattening in the early part of the graph just like the graph of these curves shows in any case, if fiat supply is growing roughly on a power of ten basis, but the bitcoin supply grows on a power of 2 basis, then yes, at a larger scale and longer timeline it's still going to flatten out, but not so soon as only 3 halving cycles in... you can literally correlate these relationships to the coordinates on this example graph
the proportionality is precisely related to the time if bitcoin is a hedge against inflation, then you would see that against the right approximate ratio of the growth rates between the two, and the proportional changes scaled vertically so the more it goes up, the less it goes up on the chart, you flatten out the variance of the exchange rate and start to see the actual adoption rate over time, and if the proportional change rate matches up closely to what is actually happening, then it will follow a linear path, time versus value should find a close to x=y ratio somewhere between the linear price and the log10 understanding this ratio correctly, and since we are coming up to the end of the fourth cycle, should give a better estimation of the likely next cycle peak, and the ones after that, and as the time continues this ratio will tend to smooth out due to the averaging of the total timescale i'm just saying, that because bitcoin's supply is power of 2, a log2 chart is the right one to use because the supply is decreasing at that rate, so its value should thus become close to linear at that scale
> i'm just saying, that because bitcoin's supply is power of 2, a log2 chart is the right one to use I’m just saying that there’s no difference, from a charting perspective because their shapes are identical, merely scaled. If you take a log10 chart, then fit it so that the min and max are at the bottom and top of the rendered height, then do the same for a log2 chart, the two charts will be literally identical, with the only exception being the labels on the y-axis. Algebraically, using a different log base results in a linear scaling factor on the output. Plotting to pixels requires its own linear scaling factor. And so a log chart will look the same regardless of the base, provided the pixel scaling factor is adjusted accordingly.
saying they are the same is like saying "on a long enough timeline the survival rate for everyone falls to zero" or that "a circle normalizes to a line", it's an a = a expression that's precisely my point... it fits a progression that has a base to it and it's somewhere around 2, for sure... it may diverge from that now but we have a long history and the divergence is not likely to be more than a standard deviation based on the correct baseline
Let’s reset. We’ll start with constant functions and work our way up to logarithms. Consider these two functions: f(x) = 1 g(x) = 2 Questions for you: A) What do these functions look like when plotted on a traditional XY Cartesian plane? B) Do they look the same or different?
you need to normalize data to its underlying scaling pattern, or you can't see the patterns in the visualisation a circle looks like a line with a big enough coefficient compared to the area under the plot, that's the point... a circle is a line, in a partial higher dimension, and same goes for two dimensional statistical data comparing two variables (time and price in this case) - patterns appear in a cluster of data when you normalize it it's not saying the graph is the normal pattern, necessarily, but if you find a good curve fit it is often predictive
> you need to normalize data to its underlying scaling pattern, or you can't see the patterns in the visualisation Correct. In a data visualization, there is a projection from the domain of the data (time, dollars, etc.) onto the range of values supported by the visual media (pixels, centimeters, etc.). Choosing the function to apply (log/linear) and parameters (base for log, offset, scale) are arbitrary and made for aesthetic reasons. > patterns appear in a cluster of data when you normalize it Agreed that the pattern you see depends on the function applied and the parameters you choose. It is my claim that the choice of base and linear scaling parameter are functionally equivalent. The remainder of this post explains how. Focusing on the Y axis, and assuming you have chosen log scale, here are the parameters you can choose: - B - base for log function - m - scale factor for linear projection - c - offset for linear projection The linear projection here is from log price to screen pixels. So the total function from price to Y coordinate is: f(p) = m * logB(p) + c Let’s consider the algebraic impact of choosing a different base, B’ for a different price projection function, f’: f’(p) = m * logB’(x) + c What is the relationship between f and f’, visually? Let’s find out. It is a rule of logarithms that one can compute a value in a new base according to this formula: logB(x) = logA(x) / logA(B) So for us, that means that: logB’(x) = logB(x) / logB(B’) Substituting this into our price projection function f: f’(p) = m * logB(x) / logB(B’) + c Refactoring: f’(p) = m * logB(x) / logB(B’) + c * logB(B’) / logB(B’) f’(p) = (m * logB(x) + c * logB(B’) ) / logB(B’) f’(p) = (m * logB(x) + c + c * (logB(B’) - 1) ) / logB(B’) Substituting our original f definition: f’(p) = (f(p) + c * (logB(B’) - 1) ) / logB(B’) Refactoring: f’(p) = f(p) / logB(B’) + c * (logB(B’) - 1) / logB(B’) Since B, B’ and c are all constants, what this last formula shows is that f’ is a linear projection of f. That is, it fits the form y=mx+b. I hope none of the above is controversial (unless I’ve made a mistake in the math). What does this mean for us, in a data visualization context? As noted earlier, visualizing a function requires projecting from the domain of the data into the range of pixel values. Putting log aside, this means, at minimum, picking a scaling factor and an offset. These values are arbitrary and chosen for aesthetic effect. So irrespective of whether we use f or f’, we’ll end up linearly scaling the values to project them into pixel space using arbitrary, aesthetically chosen parameters. If we have the same aesthetic intent in both cases, we will select projection parameters that yield identical graphs. The parameter values we pick will be different, but the pixel values will be the same (by definition, since we have the same aesthetic intent for both bases). This is what I mean when I say the graphs are identical. The only way in which they differ is by a linear scaling function, and we control arbitrary linear scaling parameters. I hope this is clear. The choice of log base and the choice of linear scaling factor are in the same category of arbitrary visualization parameters. Moreover, they have the same effect. If you squish vertically by choosing a higher base, you can stretch vertically to counterbalance that choice by choosing a larger scale factor.
