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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.

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    @Lubin there is no supposed distributivity, # is zero-order operation, before addition, and$*$is a 3rd order operation, after multiplication, it is distributive against multiplication.2012-11-06

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