in an exercise solution I was presented with the following inequality and it's not making sense to me. $ \frac{n}{n+1}\log{(\frac{1}{n}(x_1 + x_2 +...+ x_n))} +\frac{1}{n}\log{(x_{n+1})} $ $ \leq\log{(\frac{n}{n+1}(\frac{1}{n}(x_1 + x_2 +...+ x_n)) +\frac{1}{n+1}(x_{n+1})} $. It says that this inequality is due to the concavity of the log function and I see how this applies when moving $\frac{n}{n+1}$ inside the log, but I don't think it works for combining the two log terms into one. Am I missing something here that makes this a valid inequality?
2026-03-27 10:09:23.1774606163
Does the concavity of the log function justify the following inequality?
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