Jackson logarithm

This definition attempts to define the logarithm as the inverse of an exponential function. Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be regressive. Define $F \colon \mathcal{R}(\mathbb{T},\mathbb{R}) \rightarrow C_n^1(\mathbb{T},\mathbb{R})$ by $F(p)=e_p(t,s)$, where $C_n^1$ denotes nonvanishing continuously $\Delta$-differentible functions. Let $g \in C_n^1(\mathbb{T},\mathbb{R})$. Define $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$