The algebraic independence of certain transcendental numbers

Given the three numbers of, $a^\beta_1$, $a^\beta_2$, and $\frac{\ln a_2}{\ln a_1}$, where $a_1$ and $a_2$ are algebraic numbers whose logarithms are linearly independent in a rational field and $\beta$ is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.