The LIBOR market model is very popular for pricing interest rate derivatives but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term grows exponentially fast (as a function of the tenor length). We consider a Lévy-driven LIBOR model and aim to develop accurate and efficient log-Lévy approximations for the dynamics of the rates. The approximations are based on the truncation of the drift term and on Picard approximation of suitable processes. Numerical experiments for forward-rate agreements, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-Lévy approximation of annuities, which offers good approximations for high-volatility regimes.