How could I apply the Mobius inversion formulafor the equality
$$ f(x)\ln(x)+f(x/2)\ln(x/2)+f(x/3)\ln(x/3)+\dots.=g(x) \tag1$$
to get $ f(x)$ from the value of $ g(x) $ ??
The sum inside $(1)$ is infinite and $\ln$ is the natural logarithm. For the case without logarithm, I know the answer but with the logarithm I am lost.
We assume also the asymptotic $ f(x) \sim x $.