Inequalities involving logarithm

Students often ask why sometimes need to flip the inequality sign when dealing with logarithm.

This is a question asked by a student:

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For 0<x<1, both ln x and lg x are negative. When we divide by negative number, we need to change the sign.

For x>1, both ln x and lg x are positive. There is no need to change the sign when divide by ln x or lg x.

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