The properties and some applications of exponential functions over the time scales

The time scale has been a focus of interest in the world of mathematics since it was discovered. Moreover, it has a large field of application. Firstly the concept of time scale and delta differential and delta integral on the time scale, and the concept of exponential functions are defined. After then we develop a time scale logarithm that preserves certain desirable algebraic properties. In the particular, a nonlinear operator is developed that acts as an inverse for the time scale exponential in a certain sense. The scalar case is treated first, with a treatment of matrix case following. Finally we present several applications.