Restricted log-exp-analytic power functions

A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h: \mathbb{R} \to \mathbb{R}, x \mapsto \left\{\begin{array}{ll} x^r,&x>0, \\ 0,&\textnormal{ else, } \end{array}\right.$$ for $r \in \mathbb{R}$ is given. Consequently we obtain a parametric version of Tamm's theorem for this class of functions which is indeed a full generalisation of the parametric version of Tamm's theorem for $\mathbb{R}_{\textnormal{an}}^{\mathbb{R}}$-definable functions.