Premise 1: If there is no god, then logic, science and morals are not possible
Premise 2: Logic, science and morals are possible
Conclusion: Therefore god exists

The reasoning behind this argument, in the simplest of terms, is that man wouldn't be able to come up with these attributes by himself. A logic-giver, a science-giver and a moral law-giver would need to bestow these abstract attributes on us.
The apologists argue that if every person, every thinking mind, were to disappear today - logic, science and morality would still exist - it exists external from us.

There are however a few problems with this argument.
The mention of god, the very thing you hope to prove in the conclusion, being used in the first premise is circular reasoning. You cannot use god to prove god.

Secondly, TAG doesn't specify which god is being credited with dispensing these attributes. Is it a generic deist god, or a specific monotheistic version of a god?

Thirdly, reason, science and morals are certainly not derived from Christianity. The Bible, especially the Old Testament, is filled with illogical, unscientific and immoral actions. The entire Noah's Ark story, for example, is illogical, the order of creation in Genesis is unscientific and the condoning of slavery in Exodus and Deuteronomy, among other verses, is immoral.

Is god constrained by logic?
Is god also bound to work within the constraints of logic? Can god create a married bachelor? Can god draw a square circle? Can god make 1+1=3? If god is bound by logic then it is external to him as well and not derived from him. If he is bound by logic, then these attributes just exist and he too is beholden to them and cannot change them any more than we can. If god was the author of logic, science and morals he should be in a position to alter them if need be – but this seems... illogical.
Could god have made 2+2=8, or was he constrained by what was already logical?

Let's consider for a moment, if TAG were true, what would a universe look like that had no god to author these abstract attributes:
Nothing would make sense, 2+2 could equal 4 or 33 or 12 million.
A cup wouldn't just be a cup, it could also be a rainbow or a unicorn or a mountain.

What makes TAG so appealing to theists is that it answers the problem without answering the problem.
The software developer Tony Hoare said: “There are two methods in software design. One is to make the program so simple, there are obviously no errors. The other is to make it so complicated, there are no obvious errors.”
This is exactly what proponents of the TAG argument put forward. They will immediately ask 'how do you ground morality without god?' Although this question sounds profound, it is effective because it is confusing. Think about it for moment – how do you ground morality with god?

Goddunit
We may not fully understand the nature or origin of logic, science and morals, but our attempts to explain them are certainly further evolved and more comprehensive than the theistic 'god did it'.
'Goddunit' is just a theistic way of saying 'I don't know'. It doesn't add anything of value. When someone says 'God did it', he has said nothing of any substance, you don't walk away any the wiser.
'God did it' is not backed by any evidence, reason or logic. It is an 'I give-up'. It is where questioning and solution-seeking ends and apathy begins.