# Mozyrska-Torres logarithm

Let $\mathbb{T}$ be a time scale of positive numbers including $1$ and at least one other point $t$ such that $0< t < 1$. For $t \in \mathbb{T} \cap (0,\infty)$, define $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$

# Properties

Delta derivative of Mozyrska-Torres logarithm

Mozyrska-Torres logarithm at 1

Mozyrska-Torres logarithm is increasing

Mozyraska-Torres logarithm is negative on (0,1)

Mozyrska-Torres logarithm is positive on (1,infinity)

Mozyrska-Torres logarithm composed with forward jump

Euler-Cauchy logarithm

Mozyrska-Torres logarithm tends to infinity

# Special cases

Mozyrska-Torres logarithm on the reals

# See also

Bohner logarithm

Euler-Cauchy logarithm

Jackson logarithm

# References

Dorota Mozyrska and Delfim F. M. Torres: *The Natural Logarithm on Time Scales* (2008)... (next)