If $a^{(b^c)}=d^c$, find $d$ in terms of $a$ and $b$.

Question

Is it possible to express $d$ in terms of $a$ and $b$ only in the following equation? $$a^{b^c}=a^{(b^c)}=d^c$$

I want something like $d=\dots$

Thanks in advance!

Answer 1

No.

Suppose that $f(a,b)=d$ is such a function. Then $f(2,2)=4$ since for $c=1$ you get $a^{(b^1)}=2^{(2^1)}=2^2=4$. Now notice that for $c=3$ you get $2^{(2^3)}=2^8\neq 4^3=2^6$.

Answer 2

$$d=a^{b^c/c}$$ so no, because $b^c/c$ depends on $c$. For example, when $c=1$, $b^c/c=b$, when $c=-1$, then $b^c/c=-1/b$. Since $x\mapsto a^x$ is one-to-one, this means that in general, $d$ requires all three variable values as input.