Idea 3: applying a monotonic function (inequalities)

For inequalities the rule of the previous idea is refined. By applying to both sides a strictly increasing function the direction of the inequality is preserved; by applying a strictly decreasing one it is reversed. This is the reason why, with exponential and logarithmic inequalities, you must always ask yourself whether the base is greater or less than $1$.

In brief

• $a^x$ with $a>1$: increasing $\Rightarrow$ direction preserved.
• $a^x$ with $0<a<1$: decreasing $\Rightarrow$ direction reversed.
• $\log_a$ with $a>1$: increasing $\Rightarrow$ direction preserved.
• $\log_a$ with $0<a<1$: decreasing $\Rightarrow$ direction reversed.
• $t\mapsto t^2$ on $[0,+\infty)$: increasing $\Rightarrow$ you can square preserving the direction, but only if both sides are $\ge 0$.

Links

Topics: Unifying methods
Concepts: Exponential inequality · Logarithmic inequality · Monotonic function
Functions: Exponential function · Logarithmic function
Skills: Reasoning by cases · Solving inequalities