The trivial cases of the positive exponential

Since the exponential function is always strictly positive, some inequalities can be solved at a glance, without any calculation.

Warning — Trap: $a^{f(x)}<0$

An inequality of the type $a^{f(x)}<0$ is impossible (the exponential is always positive). $a^{f(x)}>0$ is, on the other hand, always true. Recognising these two trivial cases at once avoids needless mistakes.

Before launching into manipulations, it is always worth looking at the sign of the right-hand side: if it is negative, the "$>$" inequality is automatically satisfied and the "$<$" one is impossible.

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Topics: Exponential function
Concepts: Image
Functions: Exponential function
Skills: Reasoning by cases · Solving inequalities