Would a commutative power operator simplify some equations of physics? Hypertype Theory makes the radical suggestion that a commutative power operator would be preferable to the traditional non-commutative power operator $a^b$. Are there any equations in physics that would be simplified by using such an operator?
Non-commutative Power Operator: $a^b = e^{\lg(a)×b}$
Commutative Power Operator: $a \# b = b \# a = e^{\lg(a)×\lg(b)} = a^{\lg(b)} = b^{\lg(a)}$
Identity Element: $a\#e = e\#a = a$
Inverse Operator: $a \backslash b = e^{\frac{\lg(a)}{\lg(b)}} = a^{\frac{1}{\lg(b)}}$
Traditional Connection: $a^b = a\#e^b$
Terminology:
The $\#$ operation is called expansion and $a \# b$ reads “a expands b”.
The $\backslash$ operation is called contraction and $a \backslash b$ reads “a contracts b”.
Obviously, many equations like $e = mc^2$ would suffer from such a replacement and should be left untouched. The question is interested in more complex equations that could be simplified.
Note: This is important because addition and multiplication are commutative, but exponentiation is not. If one thinks of a multiplication as doing multiple additions and an exponentiation as doing multiple multiplications, the lack of commutativity for exponentiation is bothersome. Instead, one should think of addition, multiplication, and expansion as being the same operation applied to different exponential scales. Obviously, the same goes for subtraction, division, and contraction.
Algebraic Foundations: Hypertype Theory shows that an infinite number of such pairs of operators can be created in a recursive manner using very simple equations. The simplicity and beauty of the model suggests that we are looking at something interesting.
 A: I think this question gets the point of notation backwards. Good notation is invented when you notice that your calculations would look nicer if you made a few new definitions. You're trying to start with a definition you like for non-practical purposes and looking for a context where it would be practical. 
You will surely find some equations out there that are shorter with your notation, but that's true for any notation. I could define $\biguplus = 27$ and use that to shorten some equations. Feynman gives the example of defining $\mathscr{U} = |\mathbf{F} - m \mathbf{a}|^2$, so that Newton's second law simplifies to the "elegant" form
$$\boxed{\mathscr{U} = 0.}$$ 
As always, the benefit of the notation has to be traded against the mental overhead of "unpacking" it. For example, with differential forms we can boil down Maxwell's equations to 
$$d \star F = \star J.$$
This elegance is worth the overhead because we actually can compute within it; we don't have to unpack it to regular tensor notation to actually use it. But I suspect that most people who encounter your # notation will immediately unpack it to regular exponentials, because those are so much more familiar, completely negating the benefit. It's like a fancy new programming language. If you build it, they'll won't come unless it's useful.
The other roadblock, of course, is that most quantities in physics have dimensions, while your expression only makes sense if both $a$ and $b$ are dimensionless. That's why you're having more luck looking at pure math.
A: Every operator can be symmetrized, antisymmetrized, and potentially broken down into just about any combination of internal and external group structures. For your particular case, see the BCH (Baker-Campbell-Hausdorff) formula.
The operators you put forward are wrong because $a^{\log b}$ does not equal $b^{\log a}$ unless $a$ and $b$ commute to start with. You can confirm this easily by writing out the power series for $e^{\log a \log b}$ versus $e^{\log b \log a}$.
The short answer to your question is, though, that any operator which can be linearized can be put in matrix form. From there you can simply replace powers of the operation (or transcendentals in series of powers) with functions on the sum of functions of the eigenvalues.
