Question

Resolva no caderno as inequações exponenciais em IR.​

Answer 1

Caso esteja pelo app, e tenha problemas para visualizar esta resposta, experimente abrir pelo navegador https://brainly.com.br/tarefa/37401860

                                                                                                                     

$\boxed{\begin{array}{c}\sf se~a~base~\acute e~maior~que~1~\boxed{\sf o~sentido~\acute e~mantido}\\\sf se~a~base~est\acute a~entre~0~e~1~~\boxed{\sf o~sentido~se~inverte}\end{array}}$

$\tt a)~\sf 2^x<64\\\begin{array}{c|l}\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\implies 64=2^6\\\sf 2^x<2^6\\\huge\boxed{\boxed{\boxed{\boxed{\sf x<64\checkmark}}}}$

$\tt b)~\sf\dfrac{1}{16^x}\leq64\\\sf\left(\dfrac{1}{4}}\right)^{-2x}\leq64\\\sf\left(\dfrac{1}{4}\right)^{-2x}\leq\left(\dfrac{1}{4}\right)^{-3}\\\sf -2x\geq-3\cdot(-1)\\\sf 2x\leq3\\\huge\boxed{\boxed{\boxed{\boxed{\sf x\leq\dfrac{3}{2}\checkmark}}}}$

$\tt c)~\sf 0,25^{x+5}>0,5\\\sf \left[(0,5)^2\right]^{x+5}>0,5\\\sf0,5^{2x+10}>0,5^{1}\\\sf 2x+10<1\\\sf 2x<1-10\\\sf 2x<-9\\\huge\boxed{\boxed{\boxed{\boxed{\sf x<-\dfrac{9}{2}\checkmark}}}}$

$\tt d)~\sf 5^{x\cdot(3x+2)}\leq125^{x^2+4}\\\sf 5^{3x^2+2x}\leq\left[5^3\right]^{x^2+4}\\\sf 5^{3x^2+2x}\leq5^{3x^2+12}\\\sf \diagup\!\!\!\!\!3x^2+2x\leq\diagup\!\!\!\!\!3x^2+12\\\sf 2x\leq12\\\sf x\leq\dfrac{12}{2}\\\huge\boxed{\boxed{\boxed{\boxed{\sf x\leq6\checkmark}}}}$