This is, of course, a known issue, and the internet has no shortage of people raising the same concern with suggestions for a better notation. But recently, I stumbled across a math exchange post with a suggestion so lovely, so symmetrical, so utterly reasonable, that I just have to share it. For a relationship like 2 cubed equals 8, take a triangle and write 2 in the lower left, 3 on the top, and 8 on the lower right. To express the operation 2 cubed, remove that bottom right corner. The symbol as a whole represents the value that should go in the missing corner. To express log base 2 of 8, which is asking the question 2 to the what equals 8, remove the top number. To express the cube root of 8, which is saying what number to the third power equals 8, remove the bottom left corner. In other words, all three operations are completely symmetrically represented. This triangle deserves a name, and a friend of mine at Khan Academy decided that we should call it the triangle of power. The definition alone is mildly pleasing, but where it gets fun is when you see how much smoother all of the different operations become. In our current notation, there are 6 different ways to express the various inverse operations. Most of these are memorized as separate entities, some are rarely even talked about, and there's no discernible pattern even though all of them describe the same basic idea. But students still have to spend 6 times the effort to memorize each one, are 6 times more likely to make a mistake, and have 6 separate opportunities to decide math is dumb and boring and conducive to failure and why don't I just go study art instead? With the triangle of power, all these operations follow the same pattern. Our brains are really good at picking up on patterns like this, and you can much more easily imagine a smooth mental image associated with the property. There's even kind of an aesthetic pleasure to this, and who knows, maybe more of the artistically inclined students would look favorably upon math long enough to see just how valuable their intuitions really are in the science. Let's take another property, like the idea that a to the x times a to the y equals a to the x plus y. The corresponding fact for logarithms is that log of x times y equals log of x plus log of y. When you write this with the triangle of power, it's a little easier to see that both of these expressions are really saying the same thing. Remember, the symbol as a whole represents the number at the missing corner, so the top expression is saying that when you multiply two numbers that belong on the bottom right of the triangle, it corresponds with adding the numbers that belong to the top. But that's also what the lower expression is saying, when you multiply the numbers at the bottom right, it corresponds with adding numbers that belong to the top. To help students with this, you could draw inside of the triangle, saying that when the lower left is constant, the numbers at the top like to add, while the bottom right numbers like to multiply. What about when a different corner stays constant, like the top? Well in this case, you'd write a multiplication sign in both the bottom corners, because with exponents and radicals, multiplication turns into multiplication. The natural question that a student might ask from here is if there's an analogous rule for when the lower right stays constant. There is! You have to introduce a new operation, which for the sake of this video, I'll call O+, where A O plus B equals one over A plus one over B. This is not actually a ridiculous thing to introduce, since it comes up in physics all the time, like when you're computing parallel resistance. With that symbol, you could say that when the lower right number stays constant, the top numbers like to get O-plussed together, and the bottom left numbers like to get multiplied. This is actually a really nice connection between logarithms and roots, and it never gets discussed, probably because the notation isn't really conducive to asking the question.
