On intersections of the exponential and logarithmic curves

摘要

We consider the curves y = a x and y = log a x and their intersecting points for various bases a. Although this problem belongs to the elementary calculus, it turns out that the problem of determining number of these points, for a ∈ 0, 1 , is overlooked, so far. We prove that this number can be 0, 1, 2 or, even, 3, depending on the base a.