Question

Find the domain of definition of the following function:
$y=\log (\sqrt{8^{-2+\log x}}-\sqrt[3]{34^{2-\log x}}).$

Answer 1

$\sqrt{8^{-2+\log x}}-\sqrt[3]{34^{2-\log x}}>0\sqrt{8^{-2+\log x}}>\sqrt[3]{34^{2-\log x}}{\left(8^{-2+\log x}\right)}^{6/2}>{\left(4^{2-\log x}\right)}^{6/3}{\left(8^{-2+\log x}\right)}^3>{\left(4^{2-\log x}\right)}^2{8}^{-6}{8}^{3\log x}>4^4{4}^{-2\log x}{8}^{3\log x}{4}^{2\log x}>4^4{8}^6{2}^{9\log x}{2}^{4\log x}>2^8{2}^{18}{2}^{13\log x}>2^{26}13\log x>26\log x>2x>100$

Domain of $x\in (100,\infty )$