Why is $\frac{1}{\log(2x)}=\frac{1}{\log(x)}-\frac{\log 2}{\log^2(x)}+O\left ( \frac{1}{\log^3(x)} \right )$?

Question

The Development of Prime Number Theory on page 237, it says that $$ \frac{1}{\log(2x)}=\frac{1}{\log(x)}-\frac{\log 2}{\log^2(x)}+O\left ( \frac{1}{\log^3(x)} \right ) $$ At first I thought it was an expansion of $(\log(2x))^{-1}$, but WolframAlpha doesn't mention that. How did the author come to this equation?

Answer 1

Use the expansion of $\frac{1}{x+a}$ at $x=\infty$

Then plug $x\rightarrow \log(x)$ and $a\rightarrow\log 2$