Understanding the Algebraic Mistake in y = x^x Function for x < 0

[prasannapakkiam]

take the function: y = x^x
if x<0; it is not defined. However:
take this example:
x=-3
therefore: y=(-3)^(-3)
=1 / ( (-3)^3 )
=-27

HOWEVER: y == e^(xln(x))
ln(x) has a domain: {x>0, x e R}
Thus in that respect when x=-3, it is undefined.

WHY?

 

[trajan22]

I believe the identities for natural log are only valid for x>0.

 

[prasannapakkiam]

But then does taking logs on both sides technically invalid?

 

[Useful nucleus]

taking the logarithms will immediately change the domain of the function . I can give a similar example:
the domain of f(x)= sqrt[(x+1)/(x-1)] is different than the domain of g(x)=sqrt(x+1)/sqrt(x-1) . While the domain of f(x) is (1,infinty)union(-inf,-1)
, the domain of g(x) is (1,inf) .
This means certain mathematical operations may change the domain of the function like distributing the square root and taking the logarithms of both sides and it should be declared clearly the range of validity of this operation.
Unfortunately this approach is not found in any engineering Mathematics book , since these books concentrate on the applications of a theory not the theory and the accompanying definitions.