Logarithms were invented in response to the need to deal numerically and later analytically with numbers that were too big/too small for calculation abilities of that time.

Similar situation arises when dealing with tetration and higher operations today. One can try to generalize the usefulness of exp, log functions to increase/reduce order of operations by 1. We denote summation with [1], multiplication with [2], exponentiation with [3] and tetration with [4] etc.

Let us call normal logarithm 2_log, and normal exponentiation 2_exp. Then:

2_exp(a+b) = 2_exp(a[1]b)=2_exp(a)*2_exp(b) = 2_exp(a)[2]2_exp(b)

2_log (a*b) = 2_log(a)+2_log(b) = (2_log(a))[1](2_log(b))

We can form now more logarithms and exponetiations, requiring:

3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)

4_log(a[4]b)= 4_log(a)[3]4_log(b) = (4_log(a))^(4_log(b))

n_log(a[n]b)=n_log(a)[n-1] n_log(b)

This 3 log will have the same value for a^b, and b^a which would only be true for roots of equation a^b=b^a. Otherwise, 3_log must have either sign difference or index to indicate order of a and b.

Inversely, 3_exp, 4_exp and n_exp:

3_exp(a*b) =3_exp(a[2]b) = (3_exp(a))^(3_exp(b)) = 3_exp(a)[3]3_exp(b)

4_exp(a^b) = 4_exp(a[3]b)=4_exp(a)[4]4_exp(b)

n_exp(a[n-1]b)=n_exp(a)[n]n_exp(b)

Since operations above and including tetration are all too fast for numerical analysis, n_log defined in such way as above would not help much, but we could look for (3, 2) - log which reduces the order of infinity by 2 and generally for log that reduces order of infinity by as much as we need (e.g. - m). So far we stick to integer (n,m) values of orders of infinity since iteration to intermediate should be possible once the properties of functions are figured out.

(3, 2)_log (a^b) = (3, 2)_log (a[3]b)= (3,2)_log (a)+(3,2)_log (b)= (3,2)_log(a)[1](3,2)_log(b)

(4, 2)_log (a[4]b) = (4, 2)_log (a[4]b)= (4,2)_log (a)*(4,2)_log (b)= (4,2)_log(a)[2](4,2)_log(b)

In general:

(n,m)_log(a[n]b)=(n,m)_log(a) [n-m] (n,m)_log(b)

(n,m)_exp(a[n]b)=(n,m)_exp(a)[n+m] (n,m)_exp(b)

We may ask if there exists natural basis for such logarithms and exponentials, series expansion of some kind, is it related to n-factorials, and how to deal with noncommmutative/non-associative and multivalue properties of these functions arising from that.The development of the nature of such basis itself, (even if only to jump over one order of infinity, but more so over 2 or more) should be very informative.

After that we may form a 2-d table of few available values arising from (n,m) as corresponding natural basis and some values of numbers in those basis along 3rd d.

But most useful would probably be the possibility to have rules for dealing with those functions analogous to rules dealing with log ,exp based on calculus.

Which might mean most likely generalization of calculus and infinitesimal analysis-similarly how its basic form appeared soon after logarithms were invented.

Similar situation arises when dealing with tetration and higher operations today. One can try to generalize the usefulness of exp, log functions to increase/reduce order of operations by 1. We denote summation with [1], multiplication with [2], exponentiation with [3] and tetration with [4] etc.

Let us call normal logarithm 2_log, and normal exponentiation 2_exp. Then:

2_exp(a+b) = 2_exp(a[1]b)=2_exp(a)*2_exp(b) = 2_exp(a)[2]2_exp(b)

2_log (a*b) = 2_log(a)+2_log(b) = (2_log(a))[1](2_log(b))

We can form now more logarithms and exponetiations, requiring:

3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)

4_log(a[4]b)= 4_log(a)[3]4_log(b) = (4_log(a))^(4_log(b))

n_log(a[n]b)=n_log(a)[n-1] n_log(b)

This 3 log will have the same value for a^b, and b^a which would only be true for roots of equation a^b=b^a. Otherwise, 3_log must have either sign difference or index to indicate order of a and b.

Inversely, 3_exp, 4_exp and n_exp:

3_exp(a*b) =3_exp(a[2]b) = (3_exp(a))^(3_exp(b)) = 3_exp(a)[3]3_exp(b)

4_exp(a^b) = 4_exp(a[3]b)=4_exp(a)[4]4_exp(b)

n_exp(a[n-1]b)=n_exp(a)[n]n_exp(b)

Since operations above and including tetration are all too fast for numerical analysis, n_log defined in such way as above would not help much, but we could look for (3, 2) - log which reduces the order of infinity by 2 and generally for log that reduces order of infinity by as much as we need (e.g. - m). So far we stick to integer (n,m) values of orders of infinity since iteration to intermediate should be possible once the properties of functions are figured out.

(3, 2)_log (a^b) = (3, 2)_log (a[3]b)= (3,2)_log (a)+(3,2)_log (b)= (3,2)_log(a)[1](3,2)_log(b)

(4, 2)_log (a[4]b) = (4, 2)_log (a[4]b)= (4,2)_log (a)*(4,2)_log (b)= (4,2)_log(a)[2](4,2)_log(b)

In general:

(n,m)_log(a[n]b)=(n,m)_log(a) [n-m] (n,m)_log(b)

(n,m)_exp(a[n]b)=(n,m)_exp(a)[n+m] (n,m)_exp(b)

We may ask if there exists natural basis for such logarithms and exponentials, series expansion of some kind, is it related to n-factorials, and how to deal with noncommmutative/non-associative and multivalue properties of these functions arising from that.The development of the nature of such basis itself, (even if only to jump over one order of infinity, but more so over 2 or more) should be very informative.

After that we may form a 2-d table of few available values arising from (n,m) as corresponding natural basis and some values of numbers in those basis along 3rd d.

But most useful would probably be the possibility to have rules for dealing with those functions analogous to rules dealing with log ,exp based on calculus.

Which might mean most likely generalization of calculus and infinitesimal analysis-similarly how its basic form appeared soon after logarithms were invented.