[tex]\it Condi\c{\it t}iile\ de\ existen\c{\it t}\breve{a}\ a\ ecua\c{\it t}iei\ sunt:\ \ x>0,\ \ x\ne1.\\ \\ Domeniul\ de\ existen\c{\it t}\breve{a}\ este\ \ D\ =\ \mathbb{R}^*_{+}\setminus\{1\}.[/tex]
[tex]\it log_x5=\dfrac{log_55}{log_5x}=\dfrac{1}{log_5 x}\ \stackrel{not}{=}\ t,\ iar\ ecua\c{\it t}ia\ devine:\\ \\ \\ 2t^2-5t+2=0 \Rightarrow 2t^2-4t-t+2=0 \Rightarrow 2t(t-2)-(t-2)=0 \Rightarrow \\ \\ \Rightarrow (t-2)(2t-1)=0 \Rightarrow t=\dfrac{1}{2}\ sau\ t=2[/tex]
[tex]\it Revenim\ asupra\ nota\c{\it t}iei:\\ \\ t=\dfrac{1}{2} \Rightarrow log_5 x=\dfrac{1}{2} \Rightarrow x=5^{\frac{1}{2}} =\sqrt5\ \in D\\ \\ t=2 \Rightarrow log_5 x=2 \Rightarrow x=5^2=25\ \in D\\ \\ Mul\c{\it t}imea\ solu\c{\it t}iilor\ este\ S=\{\sqrt5,\ 25\}[/tex]
