In 1914 Albert Bennett suggested the following operation:
$$a * b=a^0_2b=\exp(\ln a \ln b)$$
Now, given this function, addition and multiplication, and their properties, can one express exponentiation and power function?
The operation $a^0_2b$ has the following properties:
$$a*b =b*a$$ $$(ab) * c = (a * c)(b * c)$$ $$a * e = e * a = a$$
I also wonder whether a function $a^0_{-1}b=\log(\exp(a)+\exp(b))$ can help in this endeavor.