I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so $\mathbf{E}[ln L_t]?$

I am trying to find an additive correction that would make $ln L_t$ a martingale. I am interested in the general case, but even the special case when $L_t$ has finite variation would help.