Let $x$ and $N$ be large positive numbers. Suppose that I know $x\log x\ge \log N$. How to get $x\ge C \frac{\log N}{\log\log N}$?
2026-03-30 20:50:56.1774903856
solving the log inequality $x\log x\ge \log N$
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Let me summarize Tom Chalmer's answer here: We want to prove that $x\ge \frac{\log N}{\log\log N}$. If $x>\log N$, then nothing to be proved. If $x\le\log N$, then $\log N\le x\log x\le x\log\log N$ and we are done.