Let $\Omega$ be a smooth bounded domain in ${R}^N$. We prove general uniqueness results for equations of the form $- \Delta u = au - b(x) f(u)$ in $\Omega$, subject to $u = \infty$ on $\partial \Omega$. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow-up behavior of $u(x)$ with $f(u)$ and $b(x)$, where $b \equiv 0$ on $\partial \Omega$ and a certain ratio involving $b$ is bounded near $\partial \Omega$. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov.