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Staircase

Age 16 to 18 Challenge Level:

This problem was suggested by Yatir Halevi.

Solve the equation:

$$x^{(x^3)}=3.$$

Solve the equation:

$$x^{(x^{(x^3)})}=3.$$

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.