Möbius inversion formula for two functions f(x) and g(x)

Suppose $f(x)=x^{10}$. Then $$g(x)=\sum_1^{\infty}x^{10}n^{-10}\log n=Cx^{10}$$ where $C=\sum_1^{\infty}n^{-10}\log n$ is a very small positive constant. Then $$\sum_1^{\infty}\mu(n)g(x/n)\log n=Cx^{10}\sum_1^{\infty}\mu(n)n^{-10}\log n=CDx^{10}$$ where $D=\sum_1^{\infty}\mu(n)n^{-10}\log n$ is a very small constant. We can't have $CD=1$, so we can't have $$f(x)=\sum_1^{\infty}\mu(n)g(x/n)\log n$$