Geodesic
The algebraic independence of certain transcendental numbers
Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 51-58

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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. 
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     author = {A. A. Shmelev},
     title = {The algebraic independence of certain transcendental numbers},
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A. A. Shmelev. The algebraic independence of certain transcendental numbers. Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 51-58. http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a6/