This is a table for the new mathematics notation alongside its equivalent old mathematics rendition: \begin{align} a *_1 b & \equiv a + b & & a \; /_1 \; b \equiv a - b & & a \; \backslash_1 \; b \equiv b - a \\ a *_2 b & \equiv ab & & a \; /_2 \; b \equiv \frac{a}{b} & & a \; \backslash_2 \; b \equiv \frac{b}{a} \\ a *_3 b & \equiv a^b & & a \; /_3 \; b \equiv \sqrt[b]{a} & & a \; \backslash_3 \; b \equiv \log_a b \end{align}
I chose $*$ to represent "composition" operations as I think both $+$ and $\times$ are both fairly good symbolically speaking, so I decided to merge the two. I chose $/$ and $\backslash$ to denote the "decomposition" operations as they already commonly represent division and I think they do a good job in my mind showing how these operations almost act like incisions. The subscripts correspond to the $n$th hyperoperation and continue to unfold as such.
The "forward" decomposition operations (denoted as $/$) are defined as $$ x_1 * x_2 = y \leadsto \\ y * x_2 = x_1 \leadsto \\ y = x_1 \; / \; x_2. $$
Whereas "backwards" decomposition operations (denoted as $\backslash$) are defined as $$ x_1 * x_2 = y \leadsto \\ x_1 * y = x_2 \leadsto \\ y = x_1 \; \backslash \; x_2. $$
E.g. $$ a \; /_3 \; b \equiv \sqrt[b]{a} \; \because \\ a *_3 b \equiv a^b \leadsto \\ a^b = y \leadsto \\ y^b = a \leadsto \\ y = \sqrt[b]{a} $$
and
$$ a \; \backslash_3 \; b \equiv \log_a b \; \because \\ a *_3 b \equiv a^b \leadsto \\ a^b = y \leadsto \\ a^y = b \leadsto \\ y = \log_a b $$
Hypothetically-speaking, this does imply that there should be two $*$ operations, but I don't think that would be necessarily very useful.
Are there any problems with this notation? Are there any better notations out there for hyperoperations? All I've really seen for this type of thing are Knuth's up-arrow notation and Conway's chained arrow notation, but they don't exactly seem to cover the same use cases. Please let me know if I got anything incorrect or I didn't explain something well enough.
Please note that I would have preferred to define $a \; \backslash_3 \; b$ as $\log_b a$, but it seems to have a weaker mathematical basis than the definition listed above.