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?



Irrationality measure of log(2)/log(6)
See Georges Rhin: Approximations de Padé et mesures effectives d'irrationalité. (French) [Padé approximants and effective measures of irrationality] Séminaire de Théorie des Nombres, Paris 1985–86, 155–164, Progr. Math., 71, Birkhäuser Boston, Boston, MA, 1987.


Inequality (8) there shows that if $u\_0$, $u\_1$ and $u\_2$ are integers with $H= \max(|u\_1|,|u\_2|)$ sufficiently large, then we have 
$$ 
|u\_0+u\_1\log 2 + u\_2 \log 3| \ge H^{-7.616}. 
$$ 
Thus $\eta \le 8.616$.


I don't know a free online version for Rhin's paper (you can find it on Springer Link). But you can look at [this paper](http://www.ams.org/journals/mcom/2003-72-242/S0025-5718-02-01442-4/S0025-5718-02-01442-4.pdf) by Qiang Wu in Math. Comp. which discusses similar problems and mentions Rhin's work (which seems the best known result for the $\log 2$ and $\log 3$ case). 


