We have two functions such as $y=\ln(x)$ and $y=cx^{1/2}$ and I look for the proper positive coefficient $c$ which satisfies that the graphs of the functions above intersects at only one point.
If we choose $c>1$, we see that the graphs do not intersect... If we choose $c$ between 0 and 1 we see that sometimes we obtain two intersection points and sometimes we can not find any intersection points.
Is it possible to find only one intersection point for these two functions or how can we show that it is not possible to find only one intersection point for any $c>0$ ?
If the graphs have only one intersection, they will have a common tangent (draw a diagram to see this). If the common tangent occurs at a certain value $x$ then we have $$\ln x=c\sqrt x\quad\hbox{and}\quad \frac1x=\frac{c}{2\sqrt x}\ .$$ Eliminating $x$ gives $$c=\frac2e\ .$$