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Log x is defined only for x > 0.

The first derivative of log x is 1/x, which, for x > 0 is also > 0

The second derivative of log x = -1/x2 is always negative over the valid domain for x.

Together, these derivatives show that log x is a strictly monotonic increasing function of x and that its rate of increase is always decreasing. Consequently log x is convex.

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Q: How do you show log x is convex?
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