Euler-Cauchy logarithm

Let $\mathbb{T}$ be a time scale. Define the Euler-Cauchy logarithm to be part of a solution of the Cauchy-Euler equation $$t\sigma(t)y^{\Delta \Delta}(t) - 3ty^{\Delta}(t)+4y(t)=0$$ whose linearly independent solutions are $$\left\{\begin{array}{ll} y_1(t)&=e_{\frac{2}{t}}(t,t_0) \\ y_2(t)&=e_{\frac{2}{t}}(t,t_0) \displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}. \end{array} \right.$$ This suggests that an analogue to the logarithm could be given by $$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$