  The $f(R)$ gravity and scalar-tensor theory are known to be equivalent at the
classical level. We study if this equivalence is valid at the quantum level.
There are two descriptions of the scalar-tensor theory in the Jordan and
Einstein frames. It is shown that these three formulations of the theories give
the same determinant or effective action on shell, and thus they are equivalent
at the quantum one-loop level on shell in arbitrary dimensions. We also compute
the one-loop divergence in $f(R)$ gravity.
