This problem was suggested by Yatir Halevi.

Solve the equation:

x^(x³)=3.

Solve the equation:

x^{(x^(x³))}=3.

Can you find solutions to all equations of the form:

x^{x^{x^{x^{x^{x^{...^a}}}}}}=a

where the sequence of powers is defined in the same way and a is a positive integer?

Explain what you have done and prove that you have found all possible solutions.