Need explanation on a solution of a functional equation

Question

This is a functional equation I encountered recently. I could understand the whole solution, except the part when they said that if $(2^x-k^y)(2^x-l^y)<0$ then the monotonicity is broken. Why is that? Please help me understand. Thanks in advance!

Answer 1

These are just comparing 2 different values before and after applying $g$. $g(2^{x})=2^{x},g(k^{y})=l^{y}$ by multiplicative property. So the equation is saying $(2^{x}-k^{y})(g(2^{x})-g(k^{y}))<0$ which show $g$ fail to be increasing because one of them is positive and one of them is negative, but $g$ being an increasing function require that if $2^{x}>k^{y}$ then $g(2^{x})>g(k^{y})$ and vice versa.