Large deviations for observables with logarithmic singularity and differentiability of the pressure function

We show that logarithmic observables of form $ϕ(x)=\log(|x-b|)$ on the doubling map $T$ have exponential large deviations with respect to the invariant measure if $b$ is preperiodic but not periodic with respect to $T$ giving a counterpart to the known result of strictly stretched exponential large deviations in the case that $b$ is periodic. These results are established by showing that the topological pressure function developed for unbounded observables is differentiable at $0$.