Why think of ternary relations? Well, if you do not recognize that there is an "Any two variables here determine the third" structure here, you are missing a fr

Why think of ternary relations? Well, if you do not recognize that there is an "Any two variables here determine the third" structure here, you are missing a fruitful insight into the matter. That's why. Sometimes that's a useful perspective, sometimes perhaps less so. That can be useful not only for exponentiation/logarithm/roots, incidentally. It can be useful to recognize with addition/subtraction/subtraction, or multiplication/division/division, as well. These all have a similar structure: three values in a relationship, such that any two determine the third. The triangle is supposed to help recognize this structure, though it helps with nothing else on top of that. "By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking 'it's exactly like multiplication and division, except [etc]'" The problem is, there's rather a LOT of "except". Standard * is commutative, while ^ isn't. * is associative, while ^ isn't. * turns additions on either side into additions of the result, while ^ turns multiplications on left into multiplications of the result, but multiplications on the right only into chained operations (i.e., a ^ (b * c) = (a ^ b) ^ c), while additions on the right are turned into multiplications of the result (i.e., a ^ (b + c) = (a ^ b) * (a ^ c)), and additions on the left are turned into binomial theorem expansions! These are big differences! There are particular similarities too, of course. There's a mix of similarities and lots of differences. What that amounts to pedagogically is... a muddle. Anyway, the triangle manipulations in the video aren't at all compelling to me, but regardless, the ternary relation structure IS useful to recognize (that any two out of three determine the third, and this is all the various traditional functions do), and again, this is something to be recognized not just for powers, logarithm, etc., but even for our familiar addition/subtraction or multiplication/division problems.