Nowadays the automated design of efficient floating-point implementations of correctly rounded elementary functions like cos, sin, log, exp, ... is a real challenge. Indeed, the variety of hardware architectures and floating-point formats makes such implementation process tedious and error-prone. This article focuses on the particular case of floating-point $\log_b(x)$ functions on integer processors. First it proposes a unified range reduction for $\log_b(x)$, that enables to reduce the evaluation of these functions to a single well-chosen polynomial. Second it gives some sufficient conditions on the approximation and evaluation errors to guarantee correct rounding. And third it shows how to automate the implementation process on integer processors, when $b \in \{2, \exp(1), 10\}$. Finally we illustrate how this automated approach enables to speedup the design of efficient implementations of $\log_b(x)$ for standard floating-point formats.