If we claim independence of variables for the ln multivariate normal, which looks as follows:
$-\frac{k}{2} \ln(2\pi) - \frac{1}{2} \ln|\Sigma| -\frac{1}{2}(x-\mu)\Sigma^{-1}(x-\mu)$
Since the variables are independent, the covariance is 0 except the diagonal which are the variances. So I believe we can do the following
$-\frac{k}{2} \ln(2\pi) - \frac{1}{2} \sum(\ln \,\sigma^2) -\frac{1}{2}(x-\mu)\Sigma^{-1}(x-\mu)$
Is this correct?
Also is there anything that independence can reduce from the final term:
$-\frac{1}{2}(x-\mu)\Sigma^{-1}(x-\mu)$