What we really need is not a new notation, but a notation that can be/is widely agreed on and that widely can be used.

Basicly, as the overwhelming majority of mathematical articles is written in (La)TeX (which is also an input option on this forum), we need an ASCII notation and a TeX notation.

A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is immediately going to accept it.

However there are not yet that convincing proposals for the log-type and root-type inverse functions.

I think we commonly agree on the use of slog_b for the inverse of x->b[4]x. And for ssqrt as the inverse of x->x[4]2.

One generalization would be lg[n]_b for the inverse of x->b[n]x and rt[n]^k for the inverse of x->x[n]k.

From an algebraic standpoint however our considerations fall into the category of a quasigroup. That is a set with an operation * that has unique left and right inverses. There the left and right inverses are written as \ and / respectively. This is very mnemonic as one cancels always on the corresponding side:

(a*x)/x=a and (a/x)*x=a

x\(x*a)=a and x*(x\a)=a

applied to our problem we should choose

b \[n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and

y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b.

then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2. This is even a bit mnemonic as \ reminds slightly of an l (for logarithm) and / reminds slightly of an r (for root). x/[3]k also reminds slightly of x^(1/k).

Basicly, as the overwhelming majority of mathematical articles is written in (La)TeX (which is also an input option on this forum), we need an ASCII notation and a TeX notation.

A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is immediately going to accept it.

However there are not yet that convincing proposals for the log-type and root-type inverse functions.

I think we commonly agree on the use of slog_b for the inverse of x->b[4]x. And for ssqrt as the inverse of x->x[4]2.

One generalization would be lg[n]_b for the inverse of x->b[n]x and rt[n]^k for the inverse of x->x[n]k.

From an algebraic standpoint however our considerations fall into the category of a quasigroup. That is a set with an operation * that has unique left and right inverses. There the left and right inverses are written as \ and / respectively. This is very mnemonic as one cancels always on the corresponding side:

(a*x)/x=a and (a/x)*x=a

x\(x*a)=a and x*(x\a)=a

applied to our problem we should choose

b \[n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and

y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b.

then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2. This is even a bit mnemonic as \ reminds slightly of an l (for logarithm) and / reminds slightly of an r (for root). x/[3]k also reminds slightly of x^(1/k).