Introduction

For most of us, the charts we are most familiar with are in linear space (also known as arithmetic space). Linear space is the default chart space for just about any exchange or price tracking app. This is because linear space is the chart space that essentially all people are most comfortable with and accustomed to.

You may have noticed that many exchanges and apps have an option on their charts called Log or Logarithmic Space. If you don't know what logarithmic space is or how it differs from linear space (or even what linear space is), then this post is for you.

In addition to explaining logarithmic space, I will try to convince you that log space is a critical tool for understanding the market, because it provides insights (especially on large timescales) that can not be easily observed in linear space.

Linear Space

In order to understand the shortcomings of linear space, consider the following chart.

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https://preview.redd.it/1z4u2eq2qv681.png?width=1526&format=png&auto=webp&s=7fc30c2d04069870c11039eb4cd4437a976df56a

This is a chart of the entire history of Bitcoin price since 2010 until present, in linear space. Most of you will have heard the idea that the market works in cycles, and that we are currently in the 4th such cycle. On the chart, I have pointed out the bull run portion of each of these 4 cycles.

Notice how the first bull run back in 2011 isn't even visible on the chart. The second one looks like an anthill. Even the 2017 run looks like a bunny hill compared to the current bull run, which absolutely towers over everything else.

This graph would appear to imply that the current bull run is by far the largest in history, and that they get smaller as you go back in time. This is true in terms of the absolute price (or equivalently market cap), but this can be deceiving. The reason it is deceiving is that absolute price change isn't what matters when it comes to investment growth. What matters for our investments is percentage change (or more technically: geometric change), not arithmetic price change. And in terms of percentage growth, the current bull run is by far the smallest of all the bull runs, and they get larger as you go back in time, generally speaking.

Before I go any further, let me be explicit about what linear space is: in linear space, a unit on the y-axis of a chart corresponds to some fixed quantity. On the chart in the image above, you can see that each unit equals $4,000. To put it bluntly, in linear space, the y-axis is basically a ruler.

The first bull run in 2011 saw the price of BTC go from $0.05 to about $18.30, which is a x366 increase, or a +36,500% gain. That is huge: the current bull run is so far only at x21.5 (+2,050%) measured from the previous bear market low of ~$3,220 to the recent ATH of ~$69,400.

Nevertheless, that enormous first bull run is invisible on the linear chart above, because the $18.30 ATH of that bull run is only a tiny fraction of the $4,000 that makes up a single unit in that chart's linear space.

To put it succinctly with an example: if you buy $1,000 of BTC at $1, and it then goes to $2, you make the same as if you had bought $1,000 of BTC at $30,000 and then it goes to $60,000. In both cases your $1,000 will double. However, in linear space, the jump from $30,000 to $60,000 would absolutely dwarf the jump from $1 to $2, despite the fact that these two increases represent equal gains to an investor.

What we would like is a way to compare these different market cycles in terms of actual gains; ie: where equal percentage gains are represented by equally tall hills on the chart.

This is exactly what logarithmic space is.

Logarithmic Space

In logarithmic space, units on the y-axis don't represent fixed amounts of dollars. Instead, they represent fixed multiplications (or equivalently percentage gains). For instance, there could be a logarithmic chart where each unit represents a doubling in price (+100%). There could be a logarithmic chart where each unit represents a 10x in price (+900%). Really, the multiplier that each unit represents could be anything.

The following chart depicts the same thing as the one above. The only difference is that it is logarithmic space.

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https://preview.redd.it/3je6yga5qv681.png?width=1526&format=png&auto=webp&s=155112dbfdaec9bd4f68ecaf3505886981f39f2b

**The above chart is a logarithmic chart where each unit essentially represents a doubling in price. However, TradingView has an annoying quirk where in log space, it draws those horizontal grid lines with slightly different pixel spacings, so some of those grid spaces represent multipliers that vary slightly from 2x.**

Now that we are using log space, we can easily see the 4 cycles. I have also included yellow lines to illustrate the delta between each cycle floor and the following cycle peak (for the current cycle, I just drew it to the highest peak so far. I have no idea if we will put in a higher top or not this cycle).
You can now see that the first bull market was in fact very large when it comes to % gains. Only the following run of 2013 was larger, and not by much.

This brings us to an important theory that we can now easily visualize: the theory of diminishing returns. This theory says that, generally speaking, % returns will be lesser each cyclic bull run. The only exception to this so far is the first bull run, which was slightly smaller than the second. The first bull run is often an oddball when it comes to BTC history analysis, because it happened before there had ever been a halving, and was extremely short in duration.

In any case, we can see from the chart that the bull runs have seen the following returns in chronological order: 366x, 439x, 93x, 21.5x (so far). It seems pretty clear returns are diminishing. This is in contrast to what linear space shows us, which appears to be drastically increasing returns.

Another way to understand the diminishing returns evident in this chart is this: imagine drawing a tidy curve that generally follows the overall shape of the price of BTC in the chart above. You can imagine it as an average price or curve of best fit. This curve would be a convex hill. The fact that it is convex means returns are diminishing. If returns were neither diminishing nor increasing, the curve of best fit would would a straight diagonal hill. If returns were increasing each bull market, then the curve of best fit would be a concave hill/ski jump shape.

Another theory that becomes more visible in logarithmic space is the theory of lengthening cycles. This theory basically says that each BTC cycle is longer than the previous. This is often measured from previous halving to cycle top. I didn't include the halvings in the above chart, because I didn't want to clutter it further (I will make another post that explores with greater depth into the lengthening cycle theory and the halvings). But you can see from my horizontal yellow lines that the cycles are also lengthening when measured from previous bear floor to cycle top. You can also see pretty easily from looking at this chart that the cycles generally appear to be lengthening when measured from cycle peak to next cycle peak.

In the end, the jury is still out on this theory, because nobody knows if the current cycle top is in yet. If the November 10th ATH turns out to be the cycle top, then this cycle would have been about the same length as the last one. But if this cycle drags on for a few more months before putting in a top, it will be a pretty decent corroboration of the lengthening cycles theory.

Anyway, I hope you were able to learn something from this (if you weren't learning anything then I hope you didn't read this far), and I hope you have a greater appreciation for logarithmic space, as well as an understanding of why it is so often used when doing market analysis!