I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures of logarithms of rational numbers (e.g. Wu, below).

But I wonder what the best algorithm is for comparing two such real numbers. And how does the time depend on the height of $x_1,x_2$, $y_1,y_2$, and $z_1,z_2$, in the worst case, if one wishes to know whether $$x_1 + y_1 \log(z_1) < x_2 + y_2 \log(z_2)?$$ Is something proven or conjectured here?

Wu, Qiang, On the linear independence measure of logarithms of rational numbers, Math. Comput. 72, No. 242, 901-911 (2003). ZBL1099.11037.