What is the relationship between mathematics and logic?

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It's a good idea to start with a distinction. If by "logic" you simply mean something like "correct deductive reasoning," then logic is something mathematicians use -- as do people in any discipline.

If by "logic" you mean the study of certain specific kinds of formal systems and their properties -- mathematical logic, as it's often called -- then logic is arguably a branch of mathematics, but also of philosophy (and perhaps also of other disciplines such as computer science; no need for turf wars.)

There are people in math departments who specialize in logic, and also people in philosophy departments. Results in mathematical logic, might be published in math journals or in philosophy journals or in computer science journals.

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