Mozyrska-Torres logarithm

Let $\mathbb{T}$ be a time scale. For $t \in \mathbb{T} \cap (0,\infty)$, define $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$

Properties

• $L^{\Delta}_{\mathbb{T}}(t) = \dfrac{1}{t}$
• $L_{\mathbb{T}}(1)=0$
• $L_{\mathbb{R}}(t)=\log(t)$
• $L_{\mathbb{T}}(\cdot)$ is increasing and continuous
• $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$