Exponential and logarithmic inequalities

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Solve the following inequalities:

(a) $\left(\dfrac{1}{3}\right)^{x}\le 27$;

(b) $2-3\left(\dfrac{1}{2}\right)^{x}\ge\left(\dfrac{1}{2}\right)^{x}-4$;

(c) $2^{x}+4^{x}<-1$;

(d) $\dfrac{2^{x}-2}{16-2^{x}}\ge 0$;

(e) $3^{x-1}+3^{x}\le 9^{x}$;

(f) $\dfrac{\log_2 x-1}{3-\log_2 x}\le 1$.

Solution

(a) $3^{-x}\le3^3\Rightarrow x\ge-3$.

(b) $6\ge4\left(\tfrac12\right)^x\Rightarrow\left(\tfrac12\right)^x\le\tfrac32\Rightarrow x\ge\log_{1/2}\tfrac32=-\log_2\tfrac32\approx-0.585$.

(c) $2^x+4^x>0$ always: no solution.

(d) Let $t=2^x>0$: $\tfrac{t-2}{16-t}\ge0$ for $2\le t<16\Rightarrow 1\le x<4$.

(e) $\tfrac{3^x}{3}+3^x\le3^{2x}\Rightarrow \tfrac43\le3^x\Rightarrow x\ge\log_3\tfrac43\approx0.262$.

(f) Domain $x>0$; let $u=\log_2 x$: $\tfrac{u-1}{3-u}\le1\Rightarrow\tfrac{u-2}{3-u}\le0\Rightarrow u\le2$ or $u>3\Rightarrow 0<x\le4$ or $x>8$.

$\boxed{x\ge-3;\ \ x\ge-\log_2\tfrac32;\ \ \varnothing;\ \ 1\le x<4;\ \ x\ge\log_3\tfrac43;\ \ 0<x\le4\ \lor\ x>8}$