Bohner logarithm

Define the logarithm to be $$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau$$

Note while $$\dfrac{(pq)^{\Delta}}{pq} = \dfrac{p^{\Delta}}{p} \oplus \dfrac{q^{\Delta}}{q},$$ $$\dfrac{(\frac{p}{q})^{\Delta}}{\frac{p}{q}} = \dfrac{p^{\Delta}}{p} \ominus \dfrac{q^{\Delta}}{q},$$ and $$\alpha \odot \dfrac{p^{\Delta}}{p} = \dfrac{(p^{\alpha})^{\Delta}}{p^{\alpha}}$$ the following "bad" formula holds $$L_{pq}(t,t_0)=L_p(t,t_0)+L_q(t,t_0)+\displaystyle\int_{t_0}^t \dfrac{\mu(\tau)p^{\Delta}(\tau)q^{\Delta}(\tau)}{p(\tau)q(\tau)} \Delta \tau.$$