Question

Rezolvati ecuatia 2^(3x)+1=3^(2x) , x apartine R.

Answer 1

Răspuns:

Explicație pas cu pas:

$2^{3x}+1=3^{2x}\\8^x+1=9^x |:9^x\neq0\\\left(\dfrac{8}{9}\right)^x+\left(\dfrac{1}{9}\right)^x=1\\\texttt{Consideram functia }f:\mathbb{R}\rightarrow\mathbb{R},f(x)=\left(\dfrac{8}{9}\right)^x+\left(\dfrac{1}{9}\right)^x .\texttt{Deoarece f este}\\\texttt{injectiva(fiind o suma de doua functii strict descrescatoare si}\\\texttt{prin urmare, injective), rezulta ca ecuatia f(x)=1 are cel mult}\\\texttt{o solutie.Cum f(1)=1, obtinem ca singura solutie este x=1}.\\\boxed{S=1}$