the logical hierarchy

[bo198214]

but some people always insist on

a + b

a * b

a ^ b

which is wrong.

Its not wrong its a different hierarchy.

It follows the pattern: a[n+1](b+1)=a[n](a[n+1]b)
where [n] is the the nth operation.
For example:
a[2](b+1)=a[1](a[2]b) which corresponds to a*(b+1)=a+a*b
a[3](b+1)=a[2](a[3]b) which corresponds to a^(b+1)=a*(a^b)
and so tetration [4] satisfies:
a[4](b+1)=a[3](a[4]b)