Two tangent curves

If we let $k=1/e$ , then the two graphs intersect at $(e^e,e)$ , and are also tangent at that point. A function $y=x^k$ with $0<k<1/e$ will have a value smaller than $e$ at $x=e^e$ , be greater than $y=\ln(x)$ at $x=1$ , and will at some point overtake $y=\ln(x)$ , for $k>0$ , because the derivative of $y=x^k$ is ever-increasing, whereas the derivative of $y=\ln(x)$ is ever-decreasing toward zero. Since both functions are continuous, then, by the intermediate value theorem, they must intersect twice. If $k>1/e$ , then each function value (in the relevant domain) will be greater than that of $k=1/e$ , and thus never intersect $y=\ln(x)$ .