Three exponential inequalities

Statement

Solve the following exponential inequalities:

• (a) $\left(\dfrac{1}{3}\right)^x\le 600$
• (b) $\dfrac{1}{10}-\left(\dfrac{3}{4}\right)^x\ge 0$
• (c) $2\left(\dfrac{4}{3}\right)^{x^2+x}\le 2$

Solution

(a) $\left(\tfrac13\right)^x=3^{-x}\le 600$. Taking the base-$3$ logarithm (increasing): $-x\le\log_3 600$, hence $x\ge -\log_3 600\approx -5.82$.

(b) $\left(\tfrac34\right)^x\le\tfrac{1}{10}$. The base is $<1$: the function decreases and the inequality reverses. $x\ge\log_{3/4}\tfrac{1}{10}\approx 8.00$.

(c) Divide by $2$: $\left(\tfrac43\right)^{x^2+x}\le 1=\left(\tfrac43\right)^0$. Base $>1$, hence $x^2+x\le 0\iff -1\le x\le 0$.

$\boxed{\text{(a) }x\ge -\log_3 600\approx -5.82;\quad \text{(b) }x\ge\log_{3/4}\tfrac1{10}\approx 8.00;\quad \text{(c) }-1\le x\le 0}$