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.

Challenging Inequality problem 1

Solve the inequality


From G.C, 0.5<x< 2.81, x not equal to 2 (This is the easy part)

Hence solve


This is the HOT (Higher Order Thinking) part.

Method 1

Let f(x) =


Therefore f(x+2) =


f(x+2) is translation of f(x) by 2 units in the negative X direction.

Therefore, the solution is -1.5< x<0.81, x not equal to 0

This solution combines the concept of inequality and transformation of graph.

Method 2

Replace x by x+2.

Therefore 0.5<x+2< 2.81, x+2 not equal to 2

-1.5<x<0.81, x not equal to 0