The authors study Toeplitz operators on the space \(H^{\infty}_V\) over the unit disk \(\mathbb{D}\). By definition, \(H^{\infty}_V\) consists of all analytic functions \(f\) such that, for some \(n \in \mathbb{N}\) and constant \(C_n>0\), \[ |f(z)| \leq C_n (1+|\log(1-|z|)|)^n \;\;\mathrm{for \;all} \;\;z \in \mathbb{D}. \] In the case of positive symbols, necessary and sufficient conditions for the continuity and for the compactness of Toeplitz operators are obtained in terms of growth of the Berezin transform of the defining symbol. For general, not necessarily positive symbols, sufficient conditions for the continuity and for the compactness of Toeplitz operators are also given.