A new representation of slowly varying functions

For a slowly varying function $L(t)$ a new integral representation is obtained:
$$L(t)=\eta(t)\int\limits_{t_0}^t b(x)d\ln x, t \geq t_0>0,$$
where $\eta(t)$ is measurable on $[t_0, +\infty), b(t)$ is continuous on $[t_0, + \infty)$ and $\lim\limits_{t \rightarrow + \infty} (b(t) / L(t))= 0$. This representation allows to generalize D.D. Adamovich’s classical result on equivalent slowly varying functions and to extend the statement of A. A. Goldberg theorem.