From averages with weight $1/x \log e x$ to averages with weight $1/x$

Question

Let $S:[1,\infty)\to \mathbb{R}$ be a function with $0\leq S(x)\leq 1$ for all $x$. How do I go from estimates on integrals of the form $$\int_1^w S(x)\; d \log \log e x$$to estimates on integrals of the form $$\int_1^w S(x)\; d \log x$$ (notice that the condition $|S(x)|\leq 1$ may be crucial)? Going from estimates of the second form to estimates of the first form is trivial (integration by parts). My intuition is that it should be possible to go from estimates of the first form to estimates of the second form, allowing an additional error of $+O(1)$. Is this right?