I am trying to prove that:
$$ \sum_{x} x \log(x) = \sum_{x'} x' \log(x') \implies \sum_{x} x = \sum_{x'} x' $$ for $0 \le x, x' \le 1$. I am not sure how to proceed. Can I use the monotonicity of log to prove it? TIA
2026-03-30 12:40:09.1774874409
Prove that $\sum_{x} x \log(x) = \sum_{x'} x' \log(x')$ implies $\sum_{x} x = \sum_{x'} x' $
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