I'm curious if there are any known effects of taking the element-wise logarithm of a matrix in its determinant.
so if matrix $A = \left[\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}\right]$, and $B = \left[\begin{matrix}log(a_{11}) & log(a_{12}) \\ log(a_{21}) & log(a_{22}) \end{matrix}\right]$, is there any known connection between $det(A)$ and $det(B)$?
I did some research, but so far nothing promising. I found an unanswered open question here about the effects of taking log but, but no mention of determinants. I've also googled in google scholar but it seems the vast majority of the literature deals with the logarithm of the determinant, not of the elements.
If it helps, my motivation is calculating the determinant of a matrix $A$ multiple times, but each time, its values change by an exponent, i.e. $a_{11} = x^y$ with x changing with every interaction. It would be nice to remove that frequent exponentiation. The matrix is symmetric positive definite if it helps.
If $a>0$ and$$A=\begin{bmatrix}a&a\\1/a&2/a\end{bmatrix},$$then $\det A=1$. But$$\det\left(\begin{bmatrix}\log a&\log a\\\log(1/a)&\log(2/a)\end{bmatrix}\right)=\log(2)\log(a)$$and therefore the previous determinant can take any real value.
It is easy to find a similar example in the opposite direction, that is, a family of matrices such that $\det\left[\begin{smallmatrix}\log(a_{11})&\log(a_{12})\\\log(a_{21})&\log(a_{22})\end{smallmatrix}\right]$ is constant, but $\det\left[\begin{smallmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{smallmatrix}\right]$ may take infinitely many distinct values.
I hope that this answers your question.