The logarithmic (Hencky) strain is evaluated by taking the log of the symmetric stretch tensor in continuum mechanics. Doing so requires transforming to the principal stretch basis, taking logs of the principal stretch eigenvalues, and transforming the result back to the lab basis.  While this procedure is a bit tedious, it certainly is straightforward.

The harder — almost freakishly daunting — question is: how do you get the rate of the logarithmic strain?  This rate must include contributions from both the rate of the stretch eigenvalues and the rate of the stretch eigenvectors, which is difficult to handle when there are repeated eigenvalues causing extra ambiguity of eigenvectors. 

The task of taking the derivative of a principal function (of which logarithmic strain is an example) is covered in this excerpt from Brannon’s unpublished tensor analysis book. Warning: the excerpt is taken from two parts of that book, so ignore the top part of the fourth page of the PDF, which is the trailing part of a different discussion.