The question Alternative notation for exponents, logs and roots complains that we represent strongly related concepts with vastly different notation (e.g. $x^y = z, \sqrt[y]{z} = x, \log_x z = y$) and asks if there is any alternative that would be better for pedagogical purposes.
I am wondering if there is a more general solution to simplify mathematical notation, either by reducing the number of notations or by using similar-looking notations for similar concepts (like the "triangle of power").
For example, we could do away with addition, subtraction and powers by defining $\color{blue}{\underline \phi} = \ln(\phi)$ and $\color{blue}{\overline \phi} = \exp(\phi)$:
$$\begin{align*} \color{blue}{\underline{\overline x \ \overline y \ \overline z}} & = x + y + z \\ \\ \color{blue}{x \ y \ z} &= x \cdot y \cdot z \\ \\ \color{blue}{\overline{i \pi}} &= e^{i \pi} \\ \\ \color{blue}{\overline{2 \ \underline x}} &= x^2 \\ \\ \color{blue}{\overline{- \underline 2}} &= \frac 1 2 \\ \\ \color{blue}{\overline{\overline{- \underline 2} \ \underline x} = \overline{\underline x / 2}} &= \sqrt{x} \\ \\ \end{align*}$$
Would something like this work in practice, or is there a mathematical need for the kind of motley notation we currently use?
Has anyone proposed a simpler notation such as this or a notation that is more graphically intuitive? What research has been done towards improving or standardizing mathematical notation?
