I am interested in finding the differential of the logarithm of a SPD matrix, where each component can be taken potentially in a different basis, i.e., for a diagonal matrix $D$, let $\log_\alpha(D) = \mathrm{diag}(\log_{a_1}(d_{11}),\dots, \log_{a_n}(d_{nn}))$ with $\alpha=(a_1,\dots,a_n)\in \mathbb{R}_+^n\setminus\{(1,\dots,1)\}$ and for $X=UDU^T \in S_n^{++}(\mathbb{R})$, $\log_\alpha(X) = U\log_\alpha(D)U^T$.
If $\alpha \propto (e,\dots,e)$, then we can use the Daleckii-Krein Theorem (e.g. Theorem 2.11 in https://repository.essex.ac.uk/20099/1/gendk_SIMAX_rev2c.pdf) and we find that $D_X\log(V) = U\Sigma(V)U^T$ with $\Sigma(V) = U^T V U \odot \Gamma$ with $\Gamma$ the Loewner matrix defined as $$\Gamma_{ij} = \left\{\begin{array}{ll} \frac{\log \lambda_i - \log \lambda_j}{\lambda_i - \lambda_j} & \mbox{ if }i\neq j \\ \frac{1}{\lambda_i} & \mbox{ if } i=j. \end{array}\right. $$
I was wondering if there is a similar result when using a different $f_i$ applied to each term, which would solve the problem for the differential of $\log_\alpha$? And if not, how can we derive the differential of $\log_\alpha$?