The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable fractional-type integral operators. We study convergence properties of newly constructed operators and give a quantitative form of the rate of convergence thanks to logarithmic modulus of continuity. To obtain an asymptotic formula in the sense of Voronovskaja, we consider locally regular functions. The rest of the paper devoted to approximations of newly constructed operators in logarithmic weighted space of functions. By utilizing a suitable weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem via remainder of Mellin–Taylor’s formula. Furthermore, some examples of kernels which satisfy certain assumptions are presented and the results are examined by illustrative numerical tables and graphical representations.