We consider the Edmunds--Triebel logarithmic spaces $A_\theta (logA)_{b,q} $ produced by a Banach couple $\overline{A}=(A_0, \; A_1),$ as special cases of extrapolation spaces and get estimates of a measure of weak noncompactness of the unit balls of these spaces in terms of the measures of weak noncompactness of the unit balls of the spaces $A_0$ and $A_1.$ We obtain also estimates of the $n$-th Jordan--von Neumann constant $C^n_{NJ}$ and the $n$-th James constant $J_n$ of the spaces $A_\theta (logA)_{b,q} $ in terms of the corresponding constants of the spaces $A_0$ and $A_1.$
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