Is x*(1/x)=1 propositional logic? My book says variables cannot be used in propositional logic. Is this an exception? It can't be false so I know it is a tautology, but my book doesn't categorize tautology into a type of logic.
No, this is not a formula of propositional logic.

In propositional logic, the formulas are always either propositional letters or built from smaller formulas using logical connectives. That doesn't match here because your formula is something different.

(Note that in the jargon of mathematical logic, something like $x\cdot(1/x)$ is not a "formula" but a "term". Formulas are just things that can be true or false).
This is not a propositional logic. It would be a propositional logic if it was:
$$x\neq0 \Rightarrow   x\frac{1}{x} = 1$$