The tangent to y = f(x) at the point (x0,y0) is the line through (x0,y0) with slope f'(x0): so y - f(x0) = f'(x0)(x-x0). So the tangent to y = log(x) is y-log(x0) = (1/x0)(x-x0). Similarly, the tangent to y = f(x) at the point (x1,y1) is y - f(x1) = f'(x1)(x-x1). So the tangent to y = x^2/2e is y - x1^2/2e = (x1/e)(x-x1).
You want these to be the same line. So you need the slopes and intercepts to be the same: that is, x1^2/2e = 1/x0 (slope) and log(x0)-1 = x1^2/2e - x1^2/e. You have to solve these to find x0 and x1: that gives you the equation for the common tangent.