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:

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.