“When a desperate father loses access to his only child, he attempts to take matters into his own hands, only to find himself taken hostage by his vindictive ex-wife.”

‘Access’ may be a little too vague. Does that mean he loses custody? Is there a restraining order meaning he legally isn’t allowed to see his child?
‘He attempts to take matters into his own hands’ is also vague. Take what matters into his own hands? To do what? Kill someone?
The last part is a complication in the script and doesn’t need to be included in the logline. It only raises more questions. What motive does the wife-who presumably has custody of the child- have to take him hostage? What does he do as a result of being taken hostage?

Here’s an example:?When a desperate father loses custody of his child, he must gather evidence that his ex-wife is unfit to be child’s guardian.?(23)

The inciting incident and goal should have a reciprocal relationship. The inciting incident should logically lead to the goal, and more than that, convincingly motivate it, and the goal should easily be tracked back to the inciting incident. ?The action in your logline should be specific and should draw out an idea of the plot of the story.

I hope this helps.

“Retreating from life after a tragedy, a detective on temporary leave finds out that recent murders are linked to his estranged wife and must help bring her to justice.”

What tragedy? This doesn’t even need to be in the logline, the inciting incident is when he discovers his wife is connected to murders. This needs to be more specific, how is she connected? Is she caught murdering someone in the same manner as the previous murders?
Also vague, how does he help? Why do they need him? Does he have a special skill which makes his detective skills more effective? Does he have evidence?

An example:?When his wife kills someone, a detective must use evidence from past murders to end her serial killing reign. (19)