  We consider the integrable minimal models ${\cal M}(m,m';t)$, corresponding
to the $\varphi_{1,3}$ perturbation off-criticality, in the {\it logarithmic
limit\,} $m, m'\to\infty$, $m/m'\to p/p'$ where $p, p'$ are coprime and the
limit is taken through coprime values of $m,m'$. We view these off-critical
minimal models ${\cal M}(m,m';t)$ as the continuum scaling limit of the
Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice.
Applying Corner Transfer Matrices to the Forrester-Baxter RSOS models in Regime
III, we argue that taking first the thermodynamic limit and second the {\it
logarithmic limit\,} yields off-critical logarithmic minimal models ${\cal
LM}(p,p';t)$ corresponding to the $\varphi_{1,3}$ perturbation of the critical
logarithmic minimal models ${\cal LM}(p,p')$. Specifically, in accord with the
Kyoto correspondence principle, we show that the logarithmic limit of the
one-dimensional configurational sums yields finitized quasi-rational characters
of the Kac representations of the critical logarithmic minimal models ${\cal
LM}(p,p')$. We also calculate the logarithmic limit of certain off-critical
observables ${\cal O}_{r,s}$ related to One Point Functions and show that the
associated critical exponents $\beta_{r,s}=(2-\alpha)\,\Delta_{r,s}^{p,p'}$
produce all conformal dimensions $\Delta_{r,s}^{p,p'}<{(p'-p)(9p-p')\over
4pp'}$ in the infinitely extended Kac table. The corresponding Kac labels
$(r,s)$ satisfy $(p s-p' r)^2< 8p(p'-p)$. The exponent $2-\alpha ={p'\over
2(p'-p)}$ is obtained from the logarithmic limit of the free energy giving the
conformal dimension $\Delta_t={1-\alpha\over 2-\alpha}={2p-p'\over
p'}=\Delta_{1,3}^{p,p'}$ for the perturbing field $t$. As befits a non-unitary
theory, some observables ${\cal O}_{r,s}$ diverge at criticality.
