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I completly agree with your exemple, however, I don't think you understood my argument. The big diference is that in the exemple you give, x and y have diferent value in both system, wether in my case, they have the same solution in both system. The solution of the system I proposed is, according to your argument, x=0 and y = ∞.
So both system have the same solution.
Of course, it's in general not a problem that two systems have the same solution. What is a problem is that, substituing in the same funcion the same number you sometime get 2 and sometime get 3.
In other word, if you replace in (1+x)^y x by 0 and y by ∞, do you get 2 or 3?
Would you say that the same solution is also a solution of this system? Because all your equations hold also with this system, you just have to replace you ln 2 by ln 3 and ln 0.5 by ln 1/3.
This would be very cool, because this would imply 2=3.
(In other words, I agree with anonimnystefy)
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