= Solution
The metric on covectors is induced by the inverse matrix $g^{-1}$, and on $p$-forms by the determinant pairing
$$
\langle\alpha_1\wedge\cdots\wedge\alpha_p,
\beta_1\wedge\cdots\wedge\beta_p\rangle
=\det(\langle\alpha_i,\beta_j\rangle).
$$
The <Riemannian volume form> is the unique positive top form taking value one on every oriented orthonormal frame. The <Hodge star operator> is uniquely determined by
$$
\beta\wedge *\alpha=\langle\beta,\alpha\rangle\omega_g.
$$
Nondegeneracy of the wedge pairing proves existence and uniqueness pointwise, and the smooth metric dependence makes $*$ a well-defined smooth bundle map.
On compactly supported forms, <Stokes theorem> and the graded Leibniz rule give
$$
\delta|_{\Omega^p}=(-1)^{m(p+1)+1}*d*,
$$
where $m=\dim M$; this is the formal $L^2$ adjoint of $d$. The Hodge <Laplace-Beltrami operator> is
$$
\Delta=d\delta+\delta d.
$$
For $g_f=e^{2f}g$, the covector metric scales by $e^{-2f}$, the $p$-form metric by $e^{-2pf}$, and the volume form by $e^{mf}$. Therefore
$$
*_f\alpha=e^{(m-2p)f}*\alpha.
$$
If $f$ is constant, the two star factors in the codifferential contribute $e^{-2f}$, so $\delta_f=e^{-2f}\delta$. Since $d$ is metric-independent,
$$
\Delta_f\alpha=e^{-2f}\Delta\alpha.
$$
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