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Let $X$ be an $n$-dimensional positive definite matrix and let $\log X$ denote its principal matrix logarithm. It is well-known that the following identity holds $$ \mathrm{tr}\log X = \log \det X. $$ Now, let $Y$ be another $n$-dimensional positive definite matrix and consider the following quantity $$ f(X,Y):=\mathrm{tr}(Y\log X). $$

My question. Is it possible to rewrite this function in the form $$ f(X,Y)=\log g(X,Y), $$ for some scalar function $g(X,Y)$ of $X$ and $Y$ (excluding the trivial case $g=\exp f$)? In case of negative answer, do there exist some class of matrices $Y$ for which this is possible (excluding the trivial case $Y=I$)?

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    I guess you don't just want the answer $g = e^f$? – 2017-01-29
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    @user399601: Yes, I wonder whether there exist some "less trivial" form, as in the case $Y=I$. – 2017-01-29

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