Question

va rog sa se rezolve ecuatia : lg din ⁡∛x - (lg din (x)^3)^2=0

Answer 1

   
[tex]\displaystyle\\ \text{Formule:}\\ \\ \lg \sqrt[3]{x} = \frac{\lg x}{3}\\\\ \lg x^3 = 3\lg x \\ \\ \texttt{Rezolvare} \\ \\ \lg \sqrt[3]{x} - (\lg x^3 )^2=0 \\ \\ \frac{\lg x}{3}-(3\lg x)^2=0 \\ \\ \frac{\lg x}{3}-9(\lg x)^2=0 \\ \\ \text{Substitutie: } ~\boxed{t=\lg x}\\\\ \frac{t}{3}-9t^2=0 ~~~\Big| \cdot 3\\ \\ t-27 t^2 = 0\\\\ t(1-27t)=0\\\\ t1=0 \Rightarrow \lg x = 0 \Rightarrow x_1 =10^0 =\boxed{1} [/tex]


[tex]\displaystyle\\ 1-27t=0 \Rightarrow 27t=1 \Rightarrow t_2 =\frac{1}{27}\Rightarrow \lg x= \frac{1}{27} \Rightarrow x_2 = 10^{\frac{1}{27}}= \boxed{\sqrt[27]{10}}[/tex]