The equation:
#e^y = x^x#
defines #y# as a function of #x#, but not #x# as a function of #y#.
For example:
#(1/2)^(1/2) = 1/sqrt(2) = 1/root(4)(4) = (1/4)^(1/4)#
So the equation does not determine a unique value of #x# when #y = 1/sqrt(2)# for example.
So considered as a function of #x#, what is the domain?
If #x > 0# then #x^x# is well defined and positive, so #y = ln(x^x)# is well defined.
If #x = 0# then #x^x = 0^0# is best considered an indeterminate form.
If #x < 0# then #x^x# is usually non-real complex, but does take real values, which are also positive, when #x# is an even integer. For example #(-2)^(-2) = 1/4#. So for these values #y = ln(x^x)# is also well defined.
So the domain of the function is:
#(0, oo) uu { -2, -4, -6,... } = (0, oo) uu { -2n : n in NN_1 }#
where #NN_1# denotes the positive natural numbers.
