The study of the concavity and convexity of a scalar function is an old problem studied by the mathematicians. It is perfectly known when a function is concave or convex. However, it is not so developed how to measure this concavity or convexity. If we think of a pointwise measure of convexity, as the curvature of a scalar function, the application of a concave scalar function to a convex scalar function reduces the convexity of the latter. Thus, taking into account that the logarithm is a concave scalar function, by a successive application of the logarithm to a convex scalar function at a point, we can measure the resistance that the function has to stop being convex at the point. That is, the number of times that the logarithm must be applied to the scalar function, so that it stops being convex at the point and becomes concave at it, and gives an index of pointwise measurement of the convexity of the function at the point. The degree of logarithmic convexity is defined in this way. We begin by introducing the concept of logarithmically convex function, which is a pointwise measure of the convexity of a scalar function, and by defining a few different types of convexity of the function at a point. We then extend the above concept and define an index of pointwise measurement of the convexity that we call degree of logarithmic convexity.