I think the triangle notation is actually terrible; it breaks the = symbol. A fully-filled-in triangle with all three components is essentially an equation, or a set of equations, and I can't think of any other (common) situation in which we hide the equality symbol like that. Having only two of them filled out makes me feel like it's a math problem where we're being asked to fill out the rest of the equation, not an operator.
I also don't like that this is far from the only set of operations that might fit into a triangle of some sort. In fact I've seen math problems from school using it for + and - already. I haven't seen it for * and / but it's easy to imagine. It's possible this notation is already ruined for teaching students by the common core stuff already in use. And the mere fact that the operators can be arranged in a triangle is not sufficiently unique to give the triangle to this particular set of them.
One could argue that the "=" symbol could use a rethink, but I would consider this not a terribly good place to begin that argument just because one set of operators happens to have this particular relationship.
Putting up and down arrows under exponents/roots is also not that great; it looks fine when you have one letter above the arrow but it's not going to scale well. I'd happily argue that standard exponentiation doesn't scale particularly well either once the exponents start getting complicated, but putting another symbol below it doesn't help. Putting them as inline operators flows better, but may hide the lede too much, so to speak; while the exponentiation operator we use today may have some issues, at least it's clearly visible.
Really, the problem isn't the three of exponents, roots, and log, the problem is just log. The whole "three letter operator" thing seems to have a lot of problems; see also the trig functions and their bizarre standards for sticking powers on them (where -1 is supermagical). That said, there probably isn't a problem large enough to be solvable here because the solution isn't going to be better enough to overcome inertia.
"Write a blank in an equation to mean "the thing that fills in this blank", ex."
Hmmm... I think I want something other than "a blank", but there's some promise there. I feel like your suggestion has the advantage of humbly composing with all the existing notation, whereas the triangle idea itself seems to kinda arrogantly supercede it and rewrite how equations work for just that one operator. (I've add some leading adjectives to indicate how it sort of feels to me.)
The big difference between exponentiation and addition / multiplication is that it is non commutative. This means there are two inverses. The relations between them are not very easy to learn. Using this as a teaching tool could help a lot in explaining this kind of stuff.
I remember back I high-school whenever I met these kinds of problems I would just write everything back into powers and solve the equations in that setting. Because otherwise just dealing with all the interacting and different operations was too annoying. I could see this notation essentially doing that for everyone.
Now, I made it through high-school, and got a degree in math. I think it would be better if more people made it through high school math without hating it. Making logarithms, exponents and roots clearer might help do that.
It can just be something to show to learners as a visual aid while teaching the standard notation, similar to how kids learn 10 different visual ways to add and multiply.
I also don't like that this is far from the only set of operations that might fit into a triangle of some sort. In fact I've seen math problems from school using it for + and - already. I haven't seen it for * and / but it's easy to imagine. It's possible this notation is already ruined for teaching students by the common core stuff already in use. And the mere fact that the operators can be arranged in a triangle is not sufficiently unique to give the triangle to this particular set of them.
One could argue that the "=" symbol could use a rethink, but I would consider this not a terribly good place to begin that argument just because one set of operators happens to have this particular relationship.
Putting up and down arrows under exponents/roots is also not that great; it looks fine when you have one letter above the arrow but it's not going to scale well. I'd happily argue that standard exponentiation doesn't scale particularly well either once the exponents start getting complicated, but putting another symbol below it doesn't help. Putting them as inline operators flows better, but may hide the lede too much, so to speak; while the exponentiation operator we use today may have some issues, at least it's clearly visible.
Really, the problem isn't the three of exponents, roots, and log, the problem is just log. The whole "three letter operator" thing seems to have a lot of problems; see also the trig functions and their bizarre standards for sticking powers on them (where -1 is supermagical). That said, there probably isn't a problem large enough to be solvable here because the solution isn't going to be better enough to overcome inertia.