Two exponential inequalities

Statement

Solve the inequalities, giving the critical value to two decimal places:

• (a) $3^{\,x}\le 200$
• (b) $0{.}7^{\,x}<0{.}1$

Solution

(a) $x\ln 3\le \ln 200 \Rightarrow x\le \dfrac{\ln 200}{\ln 3}=\dfrac{5{.}29832}{1{.}09861}\approx 4{.}82$. Hence $x\le 4{.}82$.

(b) Since $\ln 0{.}7<0$, dividing reverses the inequality: $x>\dfrac{\ln 0{.}1}{\ln 0{.}7}=\dfrac{-2{.}30259}{-0{.}35667}\approx 6{.}46$. Hence $x>6{.}46$.

$\boxed{x\le 4{.}82;\qquad x>6{.}46}$