I would consider the minimum distance between your functions and y=x,
The problem is as follows:
Given the two functions f(x)= e^x and f(x)=ln(x),
A.) Find the smallest possible circle that touches each function only once and all values of x and y are positive.
B.) Find the relationship between x and the radius of the circle.
Seeing how this seems like an optimization problem I wanted to see a calculus approach to this.
Since e^x and ln x are inverse functions, they're symmetric across the line x=y. So immediately you know that the center of your circle is on that line somewhere. Furthermore, you only need to minimize the distance of one of the functions to that line; the minimum distance of the other function will occur at the reflection of that point across x=y.
To minimize the distance between e^x and x=y, the first idea that comes to mind is rotating your coordinate system 45 degrees clockwise. Then x=y becomes the x-axis, and you can use calculus to minimize whatever function e^x becomes.
consider (e^x)-x. If you take its derivative, you get e^x-1. (e^x)-x has a stationary point (obvious from the graph that that is a minimum) when e^x-1=0, so at x=0. In order to find out where the circle is tangent to e^x, you could make e^x=-x, for example