Estimate about $\sum_{n\leq x}\mu(n)\log \frac xn$

Question

I'm finding the estimate of $$\sum_{n\leq x}\mu(n)\log \frac xn$$ There is a formula saying that $$\sum_{n\leq x}f(n)G\bigg(\frac xn\bigg)=\sum_{n\leq x}g(n)F\bigg(\frac xn\bigg)$$ where $F(x)=\sum_{n\leq x}f(n)$ and $G(x)=\sum_{n\leq x}g(n)$. So I want to express $\log x$ in the form $$\log x=\sum_{n\leq x}g(n)$$ for some function $g$. but what I really know is $$\log n=\sum_{d|n}\mu(d)\log\dfrac nd$$ not the same as what I want. Any suggestion? I also appreciate different way to do this.