The rule of the direction

The monotonicity of the exponential function dictates the behaviour of the direction when one “removes the base” in an inequality.

Property — Change of direction

Given an inequality $a^{f(x)} \lessgtr a^{g(x)}$:

• if $a > 1$ (increasing exponential), the direction is preserved: the inequality is equivalent to $f(x) \lessgtr g(x)$;
• if $0 < a < 1$ (decreasing exponential), the direction is reversed: it is equivalent to $f(x) \gtrless g(x)$.

Warning — Change of direction

The most frequent mistake in exponential inequalities is forgetting the change of direction when $0 < a < 1$. One way to remember it is to think of the graph: if the function is decreasing, larger values of $x$ correspond to smaller values of $y$, and vice versa — so a "$<$" on the $y$ becomes a "$>$" on the $x$.

Example — Base less than 1

Solve $\bigl(\tfrac{1}{3}\bigr)^{x-1} > \bigl(\tfrac{1}{3}\bigr)^{2x+1}$.

The base is $<1$, so the direction is reversed: $x - 1 < 2x + 1 \iff -2 < x \iff \boxed{x > -2}$.

Links

Topics: Exponential function
Concepts: Base · Monotonicity
Functions: Exponential function
Skills: Interpreting a graph · Solving inequalities