Update
This question has an answer here.
Consider the following posynomial with respect to the variables $x_1,\dots,x_n$: $$ \begin{align} f(x_1,\dots,x_n) &= \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}} \\ &= \sum_{k=1}^K c_k \cdot \prod_{i=1}^n x_i^{a_{ik}} \end{align} $$ Using the change of variables $\tilde x_i = \log x_i$, such that $x_i = e^{\tilde x_i}$, we get the following expression $$ \begin{align} f(\tilde x_1,\dots,\tilde x_n) &= \sum_{k=1}^K c_k \cdot \prod_{i=1}^n e^{\tilde x_i a_{ik}} \\ &= \sum_{k=1}^K c_k \cdot \exp\left[\sum_{i=1}^n \tilde x_i a_{ik}\right] \\ &= \sum_{k=1}^K \exp(\log(c_k)) \cdot \exp\left[\sum_{i=1}^n \tilde x_i a_{ik}\right] \\ &= \sum_{k=1}^K \exp\left[\sum_{i=1}^n \tilde x_i a_{ik} + \log(c_k)\right] \end{align} $$ This expression is convex with respect to $\tilde x_1,\dots,\tilde x_n$ (see here for a proof). My question is: does there exist a similar change of variables such that the resulting expression is concave with respect to $\tilde x_1,\dots,\tilde x_n$? That is, does there exist functions $g_1,\dots,g_n$ such that $f(\tilde x_1,\dots,\tilde x_n)$, where $\tilde x_i = g_i(x_i)$, is concave with respect to $\tilde x_1,\dots,\tilde x_n$?