The ECs always hold

A crucial clarification on the relationship between reversal of the direction and existence conditions.

Warning — The ECs hold always, even with base $<1$

The change of direction concerns only the passage from $\log_a f<\log_a g$ to $f$ vs $g$. The existence conditions ($f>0,\ g>0$) remain unchanged: they are never reversed. It is a common mistake to write "$f<0$" after having reversed the direction: no, the ECs live in a world of their own.

Links

Topics: Logarithmic function
Concepts: Existence conditions · Logarithmic inequalities · Logarithm
Functions: Logarithmic function