Abstract : During the last decade, several papers have focused on linear $q$-difference equations of the form $$ \sum_{j=0}^na_j(z)f(q^jz)=a_{n+1}(z) $$ with entire or meromorphic coefficients. A tool for studying these equations is a $q$-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order $\rho_{\log}$. It is shown, under certain assumptions, that $\rho_{\log}(f)=\max\{\rho_{\log}(a_j)\}+1$. Moreover, it is illustrated that a $q$-Casorati determinant plays a similar role in the theory of linear $q$-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the $q$-gamma function and the $q$-exponential functions all have logarithmic order two.