This essay needs a type system. Every time it says “log” it should say: log of what, into what?

It’s like audio where people say "dB" as if it answers the next question. Relative to what, measured how, and weighted for whom?

Author should brush up on https://en.wikipedia.org/wiki/Lie_theory

That's a lot of ways to think about logarithms.

Logarithms are laughably simple once you've fully internalized the meaning of the log function; it simply answers the question:

"To what power must I raise the base to get the argument?"

This is why the output tapers out as you increase the argument; because even if you increase the argument exponentially, you only need a fixed increment in the power to reach that number... So if you increase the argument only by a fixed amount (linearly) instead of exponentially, then it makes sense that the output will grow sub-linearly.

I remember when I was doing algebra with logs many years ago at school, I was applying rules to remove the log from one side of the equation.

Then when I got to uni, I had to revise the rules but it was kind of silly of me because those rules can be trivially derived if you just think about what the log function means. Turns out I had been solving equations with logs throughout school without understanding what they even meant... It's only at university that I actually bothered to learn them.

Actually TBH. I didn't even fully understand powers for some time even though I was doing calculus with them at school. I only fully understood powers once I properly internalized the concept of k-ary trees as a proxy.

It's one thing to be able to apply something, another to understand it. And I think to innovate with something, as a tool, it's not enough to be able to apply it. You must understand it.

Does this answer the question of why we see hyperoperations until exponentiation in physics, but not higher?

Logs are awesome. I started a math textbook from the 1920's a while ago, and all the calculations relied on tabulated logs, where you would convert the number to a log in a table to reduce the operation's degree, then convert back to the ordinary representation. This would reduce operations like finding cubed roots to division, would could be converted to log-log to be further reduced to subtraction before you would restore to ordinary notation. It feels like you're using a magic wormhole or something when you're doing this stuff by hand, it's really neat.