With the above notation it is immediate to check that

$$\rho^{*}e^{2(\psi-\phi_{B})}=e^{2(\rho^{*}\psi-\phi_{B^{\prime}})}.$$

We are thus reduced to a log smooth pair $(X^{\prime},B^{\prime})$, i.e. $X^{\prime}$ is smooth and $B^{\prime}$ has snc support, and the proof is then trivial. ∎