The power of 10 process can be represented by:

    10^(x/10)

The power of 2 thing can be represented by:

    (2^x)/(10^Ceiling[Log10[2^x] - 1)

This creates a curve which repeats every ~3: https://i.imgur.com/x72GsBf.png. lexicographic order is just effectively sampling what it'd look like stretching this curve to match the other curve because it doesn't repeat exactly as often as the sampling factor (i.e. it's not exactly every 3 rather every 3.0.1... meaning the samples become a different part of the curve). This stretching can more directly be represented by:

    (2^(x/s))/(10^Ceiling[Log10[2^(x/s)] - 1)

Visual with stretch set to 3.01: https://i.imgur.com/T1S8qwz.png. Visual of the differences: https://i.imgur.com/0H68ZU3.png

I'm not so sure there is any particularly strong relation beyond "you've created two curves and stretched them to be as close as possible". Would love to be proven wrong though but I think the ~ 3.01 number is the "magic" here and lexicographic order was just a way to "automatically" find that by resampling enough. There could be something as to why it so quickly matches, or it could just be how quickly the process works (in general?).