A simultaneous approximation problem for exponentials and logarithms

Let ${\alpha }_1,{\alpha }_2$ be non-zero algebraic numbers such that $\frac{\log {\alpha }_2}{\log {\alpha }_1}\mathrm{\notin }\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $(\frac{\log {\alpha }_2}{\log {\alpha }_1},{\alpha }_1^{\beta },{\alpha }_2^{\beta })$. We also establish a measure of algebraic independence of those numbers among $\frac{\log {\alpha }_2}{\log {\alpha }_1}$, ${\alpha }_1^{\beta }$ and ${\alpha }_2^{\beta }$ which are algebraically independent.