Flat (Function Attribute of associativity; removes brackets and parentheses)

Why does this function works for more than the two arguments for which it is defined?

In[100]:= log[a_ b_]:=log@a+log@b

In[101]:= log[2 x y]

Out[101]= log[2]+log[x]+log[y]

The answer is that Times has the Attribute Flat, which is Mathematica's interesting name for the mathematical property of associativity. I.e., as we learned in school, (2 + 3) + 4 = 2 + (3 + 4). We can remove the parentheses. So the Attribute is called Flat since, after the name for the function Flatten that removes List brackets, Mathematica knows that for some functions it's valid to remove all parentheses or List brackets. What is handy is to automatically remove all brackets from a user-defined function, use SetAttributes[ function, Flat].

Thus Time's Attributes show it to be more profound than we might expect. We check function Attributes like this:

In[102]:= Attributes@Plus

Out[102]= {Flat,Listable,NumericFunction,OneIdentity,Orderless,Protected}

And we can find all built-in functions with a given attribute like this, first taking all function Names from the System context where they live, using the wildcard "*", then using a pattern in Cases:

In[103]:= Names@"System`*"//Cases[#,_?(MemberQ[Attributes@#,Orderless]&)]&

Out[103]= {ArithmeticGeometricMean,BitAnd,BitOr,BitXor,CoprimeQ,DiracDelta,DiscreteDelta,Equivalent,GCD,HeavisideTheta,KroneckerDelta,LCM,Majority,Max,Min,Multinomial,Plus,Times,UnitStep,Xnor,Xor}

Last, Trace shows how the Flat Attribute affects the operation of Times in the simple example. Mathematica knows it can split up a function applied to two arguments times each other, and, typically for the evaluator, keeps applying the rule to split up all arguments of Times until there are no lists of more than two left.

In[104]:= log[2x y]//Trace

Out[104]= {log[2 x y],log[2]+log[x y],{log[x y],log[x]+log[y]},log[2]+(log[x]+log[y]),log[2]+log[x]+log[y]}