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 $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$

Properties

• $\log_{\mathbb{T}} e_p(t,s) = \dfrac{(e_p(t,s))^{\Delta}}{e_p(t,s)} = p(t)$
• If $f$ $\Delta$-differentiable nonvanishing function then $e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$
• For nonvanishing $\Delta$-differentiable functions $f,g$,

$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$

• For nonvanishing $\Delta$-differentiable functions $f,g$,

$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$