From a graphical evaluation, there is a value of c between 0.18 and 0.19 where the 2 curves are tangent, and therefore have the same slope.  For smaller values of c, the two curves intersect below the x axis.

ln(x) = cx^2

take derivative of both sides

1/x = 2cx

1 = 2cx^2

1 = 2*ln(x)

ln(x) = 1/2

x = √e

ln(√e) = c(√e)^2

1/2 = ce

c = 1/(2e) = 0.1839397 approximately

- ∞ < c <= 1/(2e)