I am new to Mathematica. So I have read the documentation for comparing expressions in Mathematica and I still don't get the SameQ (===) function. Let me give you a short example:
2+2 === 4
True
Log[x*x] === 2*Log[x]
False
I hope I'm not seeing something here. Help is much appreciated!
Every expression in Mathematica is characterized by it's FullForm.
So in your case particularly
FullForm[Log[x*x]]===Log[Power[x,2]]
and
FullForm[2*Log[x]]===Times[2,Log[x]]
These two, although equivalent from a Mathematical point of view (but only if x>0) they have different FullForm representations.
So SameQ(===) checks if the expression trees are the same.
In case more conditions are induced (like x>0) and simplifications take place then you will certainly end up to same expressions.
Assuming[x > 0, Simplify[Log[x*x]]]===2*Log[x]
returns True
1. + $MachineEpsilon === 1. and FullForm /@ {1. + $MachineEpsilon, 1.}. The tolerance is controlled by Internal`$SameQTolerance. Also, as you almost point out, for x == -1, the identity fails: {Log[(-1)^2], 2 Log[-1]} equals {0, 2 I Pi}.
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Commented
Jan 26, 2015 at 13:46
SameQ. True also about the complex solutions - We could also say that x is an object different than the usual where also the identity holds but this would be going to far :-)
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1/(a)^(3/4) === Simplify[(1/a)^(3/4), a > 0]
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