Muslims: Is recitation in Tahajjud loud or silent?

It does not matter is because Allah could hear you.

You can do it both way , as long as you do that.

Either is effective. Most people pray silently. 

See, if you believe that Allah is hearing you, whether you say it loudly or silently Allah knows that.

It was a misconception in Islam that Takbir has to be loudly said, but to be frank, it was just a recommendation from predecessors but Prophet never asked so and it was not said to do so.

The Prophet said " The best speeches are from Allah and the best actions are mine, whoever implements something new, he is not amongst us"

You should read loudly when you are an Imaam, but if you are a ma'mom or you are praying alone, it is always best to pray silently without disturbing anyone but if you are already familiar with reciting loudly then there wouldn't be anything wrong with it.

Tahajjud is a night prayer, it is part of Sunnah, so pray silently.

Pray loudly is acceptable only if you don't disturb anyone from sleeping, otherwise it is not right.

Many Imaams do not aware of this that they think turning on Qur'an recitation through audio during Fajr time loudly is okay when in fact it is form of Bidaah and creating disturbance to public. 

You wake the babies up and elders up by that. So, It is better to be a wise Muslim rather than just a Muslim alone.