clarification on logarithm problem

Question

I was wondering if someone could explain what is going on in this problem. I understand that it makes sense that $(x = 0)$
I'm not sure why $(x<0)$ or $(x>0)$ are ruled out. LS and RS mean left and right side of the (in)equality.

Answer 1

They're ruled out because you have (for example when $x<0$), 1 > LHS = RHS > 1. This is a contradiction, since we have something being simultaneously strictly greater than and strictly less than 1. The $x>0$ case follows by the same reasoning.