Let $X$ be $k \times k$ symmetric positive definite matrix and $\lambda_1,\dots, \lambda_k$ are its eigen values, then consider the transformation $Y = \text{Logm} X$ where $\text{Logm}$ is matrix logarithm.
$$J = \det \frac{dY}{dX} $$
I was reading that at one place
$$J = \left[\prod_{i=1}^{k} \frac{1}{\lambda_i} \right] \prod_{i<j} \left[ \frac{\log \lambda_i - \log \lambda_j}{\lambda_i-\lambda_j} \right] $$
and at another place it is written that
$$J = \left[\prod_{i=1}^{k} \frac{1}{\lambda_i} \right] \prod_{i<j} \left[ \frac{\log \lambda_i - \log \lambda_j}{\lambda_i-\lambda_j} \right]^2 $$
can someone correct proof and point to exact resource that explains this ?