Restricted log-exp-analytic power functions

A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$ h: \mathbb R \to \mathbb R,\quad x \mapsto \bigg \{\begin {array}{@{}ll@{}} x^r, & x \gt 0, \\ 0, & \text{else}, \end {array} $$ for $r \in \mathbb R$ is given. Consequently, a parametric version of Tamm’s theorem for this class of functions is obtained which is indeed a full generalization of the parametric version of Tamm’s theorem for $\mathbb R_{\rm an}^{\mathbb R}$-definable functions.