Jackson logarithm

Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be regressive. Let $g \colon \mathbb{T} \rightarrow \mathbb{R}$ be nonvanishing. Define the Jackson logarithm of $g$ by $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$

Properties

Theorem: The following formula holds: $$\log_{\mathbb{T}} e_p(t,s) = \dfrac{(e_p(t,s))^{\Delta}}{e_p(t,s)} = p(t).$$

Proof:

Theorem: For nonvanishing $\Delta$-differentiable functions $f,g$, $$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$$

Proof:

Theorem: If $f$ $\Delta$-differentiable nonvanishing function then $$e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$$

Proof:

Theorem: For nonvanishing $\Delta$-differentiable functions $f,g$, $$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$$

Proof: