# Log-transform of a likelihood function

I have a likelihood function $$L(\theta;\mathbf x) = \frac{\prod x_i^{\nu-1} \exp\left( -\sum x_i/\theta \right) }{\theta^{\nu n} [\Gamma(\nu)] } \qquad x>0$$

It gets log-transformed into the following formula $$\ln L(\theta;\mathbf x) = \text{constant} - \frac{n\overline x} \theta - \nu\theta\ln\theta$$

Two questions:

1. I get the same result when I perform the transformation myself, except in addition to the above result I get an extra term $$n\bar{x}(\nu-1)$$ — why is it not supposed to be there?
1. Also I get $${}-\text{const}$$ rather than $${}+\text{const}$$, but I suppose because it is an arbitrary constant value, then either $$+$$ or $$-$$ works?
• BTW, the letters $v$ and $\nu$ are not the same thing. Commented Aug 15, 2020 at 16:44
• I'm tentatively guessing that the numerator here was to be parsed as $$\left( \prod_{i=1}^n x^{\nu-1} \right) \left( \exp\left( - \sum_{i=1}^n x_i/\theta \right) \right).$$ Commented Aug 19, 2020 at 13:24
• It appears that your "extra term" should be $\displaystyle (\nu-1)\sum_i \ln x_i$ (which is a "constant" since it doesn't depend on $\theta$). $\qquad$ Commented Aug 19, 2020 at 13:27
• Also, your last term should be $\nu n\ln\theta. \qquad$ Commented Aug 19, 2020 at 13:30
• ok, My tentative guess is no longer tentative: I see where this came from. $\qquad$ Commented Aug 19, 2020 at 13:31

In this context, "constant" means not depending on $$\theta.$$ All terms that do not depend on $$\theta$$ are constant. In particular, often the next thing one does after taking the logarithm is differentiating with respect to $$\theta,$$ and then every term not depending on $$\theta$$ vanishes.