Exponential inequalities using the properties

Statement

Solve the following exponential inequalities:

• (a) $3^x\le 100$
• (b) $9\le\left(\dfrac{2}{3}\right)^x$
• (c) $-\left(\dfrac{3}{4}\right)^{x^2+1}\ge -2$

Solution

(a) Base $>1$, increasing function: $x\le\log_3 100\approx 4.19$.

(b) $\left(\tfrac23\right)^x\ge 9$. Base $<1$: taking the logarithm reverses the inequality, $x\le\log_{2/3}9\approx -5.42$.

(c) Change sign: $\left(\tfrac34\right)^{x^2+1}\le 2$. Since $x^2+1\ge 1$, the maximum of $\left(\tfrac34\right)^{x^2+1}$ occurs at $x=0$ and equals $\tfrac34<2$: the inequality is always satisfied, $\forall x\in\mathbb{R}$.

$\boxed{\text{(a) }x\le\log_3 100\approx 4.19;\quad \text{(b) }x\le\log_{2/3}9\approx -5.42;\quad \text{(c) }\forall x\in\mathbb{R}}$