The necessary condition for to have an inflection point is that there exists a in so that . A sufficient condition for to have an inflection point is that changes signs at .
First, let us simplify the function . Using the change of base of logarithms formula given by
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Take , , and , we obtain that

Thus
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Now we take the first and second derivatives of . Recall that if , then . Taking the first derivative of , we get:
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Now, we take the second derivative of and find:
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Remark next that is a continuous function of on the interval , since both and are continuous functions of on . Now we want to show that there exists a so that . Evaluate gives:
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Now evaluate to get:
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Since is continuous on , and while , by the Intermediate Value Theorem, there exists a point so that . We have already shown that there is a point to the left and to the right of c with different signs. So indeed the given function has an inflection point on .
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