Some days ago, I did a puzzle using exp(x) and ln(x) and it occurred to me that these functions can be used to start off a different type of puzzle.

Let x = 1. Then you have the points (1,e) and (1,0). Now draw tangents to the curves at those points. The tangent lines (most likely - I haven't formally figured it out this far) would intersect at a point. Figure out what this point is. Now let x = 2 to get the points (2, exp(2)) and (2, ln(2)). Now draw tangents to those two points and figure out what the point of intersection is for the next tangent lines. Let x take on various different values and derive more points of intersection.

Can you produce a graph showing the locus of intersection points? As an extra bonus question, can you derive an equation for these locus points? (to take it a step further, you may want to explore what happens letting y = x +1 or using parabolas for variety sakes).