Let X be a positive definite matrix.

$\mbox{det} X=\mbox{exp} \mbox{tr} \mbox{ln} X$

or

$\mbox{ln} \mbox{det} X=\mbox{tr} \mbox{ln }X$

See here for application.

EDIT: very useful, especially for MLE of Gaussian parameters. In addition,

$\mathrm{d}\ln A=A^{-1}\mathrm{d}A$

$\mathrm{d}\ln |A|=\mathrm{tr}(A^{-1}\mathrm{d}A)A$

Because: $\mathrm{d}\ln |A|=\mathrm{d}\mathrm{tr}\ln A=\mathrm{tr}(\mathrm{d}\ln A)=\mathrm{tr}A^{-1}\mathrm{d}A$