Guided examples with base less than 1

Five guided examples with base less than $1$. The rule is always the same: the direction is reversed. Let us see the cases most frequent in written papers.

Example — Base $1/2$, reduced to the same base

$\bigl(\tfrac{1}{2}\bigr)^{2x+1}\le \bigl(\tfrac{1}{2}\bigr)^{x-3}$. Base $<1\Rightarrow$ direction reversed: $2x+1\ge x-3 \iff x\ge -4$. Solution: $\boxed{x\ge -4}$.

Example — Rewriting with base less than 1

$5^{x+1}>\bigl(\tfrac{1}{5}\bigr)^{x-2}$. Rewriting $\bigl(\tfrac{1}{5}\bigr)^{x-2}=5^{-(x-2)}=5^{2-x}$: $5^{x+1}>5^{2-x} \iff x+1>2-x \iff x>\tfrac{1}{2}.$

Example — Decimal base $1/10$

$\bigl(\tfrac{1}{10}\bigr)^{x^2-1}<\bigl(\tfrac{1}{10}\bigr)^{2x-1}$. Base $<1$: $x^2-1>2x-1 \iff x^2-2x>0\iff x(x-2)>0 \iff x<0\ \vee\ x>2$.

Example — Substitution with base less than 1

$\bigl(\tfrac{1}{2}\bigr)^{2x}-5\bigl(\tfrac{1}{2}\bigr)^x+4\ge 0$. I set $t=(1/2)^x>0$: $t^2-5t+4\ge 0 \iff (t-1)(t-4)\ge 0 \iff t\le 1 \ \vee\ t\ge 4.$ I return to $x$: $(1/2)^x\le 1 \iff x\ge 0$ (direction reversed, base $<1$); $(1/2)^x\ge 4=(1/2)^{-2}\iff x\le -2$. Solution: $\boxed{x\le -2 \ \vee\ x\ge 0}$.

Example — The "trick" of comparing with $1$

$\bigl(\tfrac{2}{3}\bigr)^{x^2-4}\ge 1$. I rewrite $1=(2/3)^0$: $\bigl(\tfrac{2}{3}\bigr)^{x^2-4}\ge\bigl(\tfrac{2}{3}\bigr)^0$. Base $<1$: $x^2-4\le 0 \iff -2\le x\le 2$.

Links

Topics: Exponential function
Concepts: Base · Monotonicity
Functions: Exponential function
Methods: Exponential equations
Skills: Reasoning by cases · Solving inequalities