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Log

Let

$\displaystyle z_i=\log(x_i), \; 1\leq i \leq N.$

Then, the Jacobian is

$\displaystyle \frac{\partial z_i}{\partial x_i} = \frac{1}{x_i},$

so

$\displaystyle \log J = - \sum_{i=1}^N \log x_i = -\sum_{i=1}^N z_i.$

For more information, refer to the software module software/module_log.m.