Let $A \asymp B$ mean that there exists universal constants $m,M >0$ such that $mA \leq B \leq MA$. Let $k,n \in \mathbb{N}$ be such that $\log n \leq k \leq n$. I want to prove that
$$ k \log(\frac{n}{k}+1)=m \Rightarrow k \asymp \frac{m}{\log(\frac{n}{m}+1)}. $$ Can anyone give me useful directions about where to start with? I am finding it difficult to express $k$ as a function of $m$ explicitly.