Question

Solve inequality ${\log }_3({16}^x-2\left({12}^x\right))\le 2x+1$

Answer 1

${\log }_3({16}^x-2({12}^x\left)\right)\le 2x+1$

$\Rightarrow {16}^x-2\left({12}^x\right)\le 3^{2x+1}$

$\Rightarrow 4^{2x}-2(3^x\times 4^x)\le {3.3}^{2x}$

$\Rightarrow \frac{4^{2x}-2(3^x\times 4^x)}{{3.3}^{2x}}\le 1$

$\Rightarrow \frac{4^{2x}}{{3.3}^{2x}}-\frac{2(3^x\times 4^x)}{{3.3}^{2x}}\le 1$

$\Rightarrow \frac{1}{3}(\frac{4}{3}{)}^{2x}-\frac{2}{3}(\frac{4}{3}{)}^x\le 1$

Let $(\frac{4}{3}{)}^x=t$

$\Rightarrow \frac{1}{3}{t}^2-\frac{2}{3}t\le 1$

$\Rightarrow t^2-2t\le 3$

$\Rightarrow t^2-2t-3\le 0$

$\Rightarrow (t-3)(t+1)\le 0$

$\Rightarrow -1\le t\le 3$

Since $(\frac{4}{3}{)}^x>0$

$\Rightarrow 0<t\le 3$

Therefore $0<(\frac{4}{3}{)}^x\le 3$