Exists $f$ such that $\int |f|<+\infty$ but $\int |f\cdot\log x|=\infty$?

Question

I read that $(L^1)'=L^\infty$ (dual space). This means that $\log(x)1_{[1,+\infty]}(x)$ doesn't belong to $(L^1)'$ since $\log$ is unlimited. Hence, it should exist $f\in L^1$ s.t. $$\int_1^\infty |f(x)\log(x)|dx=+\infty$$ But I cannot find that function

Answer 1

What about

$$f(x)=\frac1{x(1+\log^2 x)}$$

in this case indeed

$$\int_1^\infty f(x) dx<\infty$$

but

$$\int_1^\infty f(x)\log(x) dx=\infty$$

Answer 2

Try something like the function $$f(x)=\frac{1}{x(\ln^2(x)+1)}$$