Transforming a power tower to a product

Question

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and $g=\log$, since $$\exp\left(\log\prod_i a_i\right)=\exp\left(\sum_i \log a_i\right)$$ Is there a similar way to write the power tower of a sequence of terms as $$a_0^{a_1^{a_2^{a_3^\ldots}}}=f\left(\prod_i g(a_i)\right)$$

Answer 1

No. To see this, observe that the RHS of your expression remains the same if I swap $a_0$ and $a_1$, but the LHS does not.