Are the notations used for exponents and logarithms confusing?

We express 2*2*2 as 2 superscript 3, which can also be expressed as 2^3. I have two objections to this. Firstly, the superscript notation does not have a symbol for an operator, though the caret symbol introduces one. Additionally, placing the 3 after the 2 is putting the cart before the horse. We are multiplying 2 three times. Wouldn’t it make more sense to express this as 3^2?

Consider multiplication. We express 2+2+2 as 3*2, which seems quite natural. It makes it easy to understand that 3x + 2x is the combination of 3 x’s and 2 x’s, giving 5 x’s, or 5x.

Similarly, if 2*2*2 is expressed as 3^2, then it makes it easier to understand that 3^2 * 5^2 = 8^2. I have seen students struggling to remember the exponent addition rule, which is something that should come naturally.

The notation for logarithms is equally confusing. log subscript 2 of 8 = 3. Suppose that instead we expressed this as 2 v 8 = 3. Now watch what happens when we combine operations. We get 2 ^ (2 v 8) = 8. 2 v (2 ^ 3) = 3. The operations cancel, because they are inverses of each other, which is something that most students never come to understand.