As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity

$\eta = \inf \{\alpha : \left|\frac{\log 2}{\log 6} - \frac{p}{q}\right| < \frac{1}{q^\alpha} \text{ for only finitely many $p,q\in\mathbf Z$}\}$.

I have found a paper *An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers* by E.M. Matveev. This paper gives a complicated system of parameters which can be adjusted bound the irrationality measure of rational linear combinations of logs. From this I have managed to obtain the bound

$\eta < 10^9$.

Does anyone know of references that can produce a better bound on this quanity? Perhaps one closer to its likely value of 2?