\ln t_s = \ln (t-t_{N-1}) - \sum_{i=1}^{N-1} frac{q_i - q_{i-1}}{q_{N-1}}  \ln (t-t_{i-1}) = \ln \Delta t - \sum_{i=1}^{N-1} \frac{q_i - q_{i-1}}{q_{N-1}}  \ln \left( \Delta t + \sum_{i=1}^{N-i} \Delta t_i \right)