Minorant principle for function $\frac{\ln(x)} x$

Question

I am trying to use minorant principle on:

$$\int_1^\infty\frac{\ln(x)} x \, dx$$

It's easy enough to integrate and figure out that the integral diverges, but is there a function that is smaller than the this and for which the sum diverges clearly?

I am trying to avoid integration as best as I can.