In this talk, we discuss about the regularity properties of fractional logarithmic p-Laplacian. We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic p-Laplace operator. For a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order s\in (0, 1). The kernel of this operator involves a logarithmic factor and it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional p-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in our setting. We also present an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear case p=2.