Clearly it's not arbitrary when LookIntoBitcoin have modified the logarithmic curve, and haven't referenced the fact it's arbitrary or lacks any scientific basis. Probably because it's a mathematical based curve, nothing to do with science. The irony being that a logarithmic curve in itself isn't arbitrary, it's simply how you style it (connect the dots), even if the original remains arbitrary and unchanged.
You didn't get me. The Log curve is an approximation of price history (logarithmic regression). That is the scientific part. But the rainbow is made by duplicating that curve parallel to itself with certain intervals that are not somehow "scientifically" linked to volatility, ATHs ant Lows.
And it does not look like they have modified the curve either. Here is my view of the trend as for now

Fair enough I hear ya. Looks like blockchaincenter have had a botched version for sometime now then, as clearly
it's very different than how it should be. Thanks for pointing out which one is more accurate, I'll stick to the lookintobitcoin version from now on, even if it's not clear if this is based on the original model:
The first example of a logarithmic regression curve for Bitcoin was created by Bitcoin Talk user trolololo. This is the original chart, which shows how $BTC price could potentially evolve over time using log growth regression analysis:
It seems they are almost implying this is a different version of the rainbow chart (no copyright) that's based on logarithmic values, but not necessarily the "original hard-coded version". If you look more in-depth, I think you'll find it slightly different (look at the March 2020 lows as well as current price), because ultimately they've made a good effort to remove any confirmation that they using the original arbitrary version.