Difference between revisions of "Jackson logarithm"

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=Properties=
 
=Properties=
 
[[Jackson logarithm of delta exponential]]<br />
 
[[Jackson logarithm of delta exponential]]<br />
 
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[[Delta exponential of Jackson logarithm]]<br />
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<strong>Theorem:</strong> For nonvanishing $\Delta$-differentiable functions $f,g$,
 
$$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> If $f$ $\Delta$-differentiable nonvanishing function then
 
$$e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> For nonvanishing $\Delta$-differentiable functions $f,g$,
 
$$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$$
 
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<strong>Proof:</strong> █
 
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=See also=
 
=See also=

Revision as of 17:34, 11 February 2017

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

Jackson logarithm of delta exponential
Delta exponential of Jackson logarithm

See also

Bohner logarithm
Euler-Cauchy logarithm
Mozyrska-Torres logarithm

References