Calculate this power?

Question

Let $a$ is a positive real number. If ${a^{{a^{{a^{16}}}}}} = 16$ how much is ${a^{{a^{{a^{12}}}}}}$?

Answer 1

Define the function $f(x)=a^x$

The equation $$a\uparrow a\uparrow a\uparrow 16=16$$

can be written in the form $$f(f(f(16)))=16$$

Obviously, the solution of the equation $f(16)=16$ solves this equation. Because of the monotony of $f(f(f(16)))-16$, it is the only solution, so we have $$a^{16}=16$$

Therefore, we have $$a^4=2$$ or $$a^{12}=8$$

Furthermore, we have $$a^{a^{12}}=a^8=4$$

and finally $$a^{a^{a^{12}}}=a^4=2$$